Practice · Two equal inputs break a function · Sensim Math

Variant 1 easy

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$2$	$m-1$	$5$	$2m+1$
$y$	$2$	$3$	$4$	$5$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $2$ , $3$ , $4$ , $5$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $2$ , $m-1$ , $5$ , $2m+1$ .
The matching $y$ entries are $2$ , $3$ , $4$ , $5$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

2

and

5

can never be equal, which leaves five pairs that involve

m

.

m-1=2,\quad 2m+1=2,\quad m-1=5,\quad 2m+1=5,\quad m-1=2m+1

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=3,\quad m=\frac{1}{2},\quad m=6,\quad m=2,\quad m=-2

For example

2m+1=2

gives

2m=1

, so

m=\frac{1}{2}

, and

m-1=2m+1

gives

m=-2

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

\frac{1}{2}

,

2

,

3

,

6

4 · Reviewdoes it hold up?

Check $m=-2$ : the $x$ entries are $2$ , $-3$ , $5$ , $-3$ , so $x=-3$ goes to both $3$ and $5$ . Check $m=2$ : the entries are $2$ , $1$ , $5$ , $5$ , so $x=5$ goes to both $4$ and $5$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $2$ , $-1$ , $5$ , $1$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 2 easy

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$6$	$2m-3$	$2$	$m+1$
$y$	$1$	$2$	$3$	$4$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $1$ , $2$ , $3$ , $4$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $6$ , $2m-3$ , $2$ , $m+1$ .
The matching $y$ entries are $1$ , $2$ , $3$ , $4$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

6

and

2

can never be equal, which leaves five pairs that involve

m

.

2m-3=6,\quad m+1=6,\quad 2m-3=2,\quad m+1=2,\quad 2m-3=m+1

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=4\frac{1}{2},\quad m=5,\quad m=2\frac{1}{2},\quad m=1,\quad m=4

For example

2m-3=6

gives

2m=9

, so

m=4\frac{1}{2}

, and

2m-3=m+1

gives

m=4

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

1

,

2\frac{1}{2}

,

4

,

4\frac{1}{2}

,

5

4 · Reviewdoes it hold up?

Check $m=4$ : the $x$ entries are $6$ , $5$ , $2$ , $5$ , so $x=5$ goes to both $2$ and $4$ . Check $m=1$ : the entries are $6$ , $-1$ , $2$ , $2$ , so $x=2$ goes to both $3$ and $4$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $6$ , $-3$ , $2$ , $1$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 3 easy

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$2$	$m+4$	$6$	$3m-2$
$y$	$0$	$1$	$2$	$3$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $0$ , $1$ , $2$ , $3$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $2$ , $m+4$ , $6$ , $3m-2$ .
The matching $y$ entries are $0$ , $1$ , $2$ , $3$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

2

and

6

can never be equal, which leaves five pairs that involve

m

.

m+4=2,\quad 3m-2=2,\quad m+4=6,\quad 3m-2=6,\quad m+4=3m-2

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=-2,\quad m=1\frac{1}{3},\quad m=2,\quad m=2\frac{2}{3},\quad m=3

For example

3m-2=2

gives

3m=4

, so

m=1\frac{1}{3}

, and

m+4=3m-2

gives

m=3

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

1\frac{1}{3}

,

2

,

2\frac{2}{3}

,

3

4 · Reviewdoes it hold up?

Check $m=3$ : the $x$ entries are $2$ , $7$ , $6$ , $7$ , so $x=7$ goes to both $1$ and $3$ . Check $m=2\frac{2}{3}$ : the entries are $2$ , $6\frac{2}{3}$ , $6$ , $6$ , so $x=6$ goes to both $2$ and $3$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $2$ , $4$ , $6$ , $-2$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 4 easy

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$0$	$2m-1$	$3$	$m+2$
$y$	$3$	$4$	$5$	$6$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $3$ , $4$ , $5$ , $6$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $0$ , $2m-1$ , $3$ , $m+2$ .
The matching $y$ entries are $3$ , $4$ , $5$ , $6$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

0

and

3

can never be equal, which leaves five pairs that involve

m

.

2m-1=0,\quad m+2=0,\quad 2m-1=3,\quad m+2=3,\quad 2m-1=m+2

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=\frac{1}{2},\quad m=-2,\quad m=2,\quad m=1,\quad m=3

For example

2m-1=0

gives

2m=1

, so

m=\frac{1}{2}

, and

2m-1=m+2

gives

m=3

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

\frac{1}{2}

,

1

,

2

,

3

4 · Reviewdoes it hold up?

Check $m=3$ : the $x$ entries are $0$ , $5$ , $3$ , $5$ , so $x=5$ goes to both $4$ and $6$ . Check $m=1$ : the entries are $0$ , $1$ , $3$ , $3$ , so $x=3$ goes to both $5$ and $6$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $0$ , $-1$ , $3$ , $2$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 5 medium

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$-1$	$m+1$	$3$	$2m-3$
$y$	$1$	$3$	$5$	$7$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $1$ , $3$ , $5$ , $7$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $-1$ , $m+1$ , $3$ , $2m-3$ .
The matching $y$ entries are $1$ , $3$ , $5$ , $7$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

-1

and

3

can never be equal, which leaves five pairs that involve

m

.

m+1=-1,\quad 2m-3=-1,\quad m+1=3,\quad 2m-3=3,\quad m+1=2m-3

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=-2,\quad m=1,\quad m=2,\quad m=3,\quad m=4

For example

2m-3=-1

gives

2m=2

, so

m=1

, and

m+1=2m-3

gives

m=4

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

1

,

2

,

3

,

4

4 · Reviewdoes it hold up?

Check $m=4$ : the $x$ entries are $-1$ , $5$ , $3$ , $5$ , so $x=5$ goes to both $3$ and $7$ . Check $m=3$ : the entries are $-1$ , $4$ , $3$ , $3$ , so $x=3$ goes to both $5$ and $7$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $-1$ , $1$ , $3$ , $-3$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 6 medium

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$3$	$m+2$	$7$	$2m-1$
$y$	$1$	$2$	$3$	$4$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $1$ , $2$ , $3$ , $4$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $3$ , $m+2$ , $7$ , $2m-1$ .
The matching $y$ entries are $1$ , $2$ , $3$ , $4$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

3

and

7

can never be equal, which leaves five pairs that involve

m

.

m+2=3,\quad 2m-1=3,\quad m+2=7,\quad 2m-1=7,\quad m+2=2m-1

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=1,\quad m=2,\quad m=5,\quad m=4,\quad m=3

For example

2m-1=3

gives

2m=4

, so

m=2

, and

m+2=2m-1

gives

m=3

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

1

,

2

,

3

,

4

,

5

4 · Reviewdoes it hold up?

Check $m=3$ : the $x$ entries are $3$ , $5$ , $7$ , $5$ , so $x=5$ goes to both $2$ and $4$ . Check $m=4$ : the entries are $3$ , $6$ , $7$ , $7$ , so $x=7$ goes to both $3$ and $4$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $3$ , $2$ , $7$ , $-1$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 7 medium

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$0$	$m-3$	$1$	$2m-1$
$y$	$4$	$5$	$6$	$7$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $4$ , $5$ , $6$ , $7$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $0$ , $m-3$ , $1$ , $2m-1$ .
The matching $y$ entries are $4$ , $5$ , $6$ , $7$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

0

and

1

can never be equal, which leaves five pairs that involve

m

.

m-3=0,\quad 2m-1=0,\quad m-3=1,\quad 2m-1=1,\quad m-3=2m-1

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=3,\quad m=\frac{1}{2},\quad m=4,\quad m=1,\quad m=-2

For example

2m-1=0

gives

2m=1

, so

m=\frac{1}{2}

, and

m-3=2m-1

gives

m=-2

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

\frac{1}{2}

,

1

,

3

,

4

4 · Reviewdoes it hold up?

Check $m=-2$ : the $x$ entries are $0$ , $-5$ , $1$ , $-5$ , so $x=-5$ goes to both $5$ and $7$ . Check $m=1$ : the entries are $0$ , $-2$ , $1$ , $1$ , so $x=1$ goes to both $6$ and $7$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $0$ , $-3$ , $1$ , $-1$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 8 medium

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$1$	$m-3$	$4$	$3m+1$
$y$	$2$	$4$	$6$	$8$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $2$ , $4$ , $6$ , $8$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $1$ , $m-3$ , $4$ , $3m+1$ .
The matching $y$ entries are $2$ , $4$ , $6$ , $8$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

1

and

4

can never be equal, which leaves five pairs that involve

m

.

m-3=1,\quad 3m+1=1,\quad m-3=4,\quad 3m+1=4,\quad m-3=3m+1

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=4,\quad m=0,\quad m=7,\quad m=1,\quad m=-2

For example

3m+1=1

gives

3m=0

, so

m=0

, and

m-3=3m+1

gives

m=-2

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-2

,

0

,

1

,

4

,

7

4 · Reviewdoes it hold up?

Check $m=-2$ : the $x$ entries are $1$ , $-5$ , $4$ , $-5$ , so $x=-5$ goes to both $4$ and $8$ . Check $m=1$ : the entries are $1$ , $-2$ , $4$ , $4$ , so $x=4$ goes to both $6$ and $8$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=-1$ , the entries $1$ , $-4$ , $4$ , $-2$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 9 hard

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$5$	$m-2$	$1$	$2m+3$
$y$	$2$	$4$	$6$	$8$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $2$ , $4$ , $6$ , $8$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $5$ , $m-2$ , $1$ , $2m+3$ .
The matching $y$ entries are $2$ , $4$ , $6$ , $8$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

5

and

1

can never be equal, which leaves five pairs that involve

m

.

m-2=5,\quad 2m+3=5,\quad m-2=1,\quad 2m+3=1,\quad m-2=2m+3

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=7,\quad m=1,\quad m=3,\quad m=-1,\quad m=-5

For example

2m+3=5

gives

2m=2

, so

m=1

, and

m-2=2m+3

gives

m=-5

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-5

,

-1

,

1

,

3

,

7

4 · Reviewdoes it hold up?

Check $m=-5$ : the $x$ entries are $5$ , $-7$ , $1$ , $-7$ , so $x=-7$ goes to both $4$ and $8$ . Check $m=-1$ : the entries are $5$ , $-3$ , $1$ , $1$ , so $x=1$ goes to both $6$ and $8$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $5$ , $-2$ , $1$ , $3$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 10 hard

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$4$	$2m+1$	$6$	$m+3$
$y$	$5$	$6$	$7$	$8$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $5$ , $6$ , $7$ , $8$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $4$ , $2m+1$ , $6$ , $m+3$ .
The matching $y$ entries are $5$ , $6$ , $7$ , $8$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

4

and

6

can never be equal, which leaves five pairs that involve

m

.

2m+1=4,\quad m+3=4,\quad 2m+1=6,\quad m+3=6,\quad 2m+1=m+3

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=1\frac{1}{2},\quad m=1,\quad m=2\frac{1}{2},\quad m=3,\quad m=2

For example

2m+1=4

gives

2m=3

, so

m=1\frac{1}{2}

, and

2m+1=m+3

gives

m=2

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

1

,

1\frac{1}{2}

,

2

,

2\frac{1}{2}

,

3

4 · Reviewdoes it hold up?

Check $m=2$ : the $x$ entries are $4$ , $5$ , $6$ , $5$ , so $x=5$ goes to both $6$ and $8$ . Check $m=3$ : the entries are $4$ , $7$ , $6$ , $6$ , so $x=6$ goes to both $7$ and $8$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $4$ , $1$ , $6$ , $3$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 11 hard

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$-2$	$m-4$	$4$	$3m+2$
$y$	$3$	$6$	$9$	$12$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $3$ , $6$ , $9$ , $12$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $-2$ , $m-4$ , $4$ , $3m+2$ .
The matching $y$ entries are $3$ , $6$ , $9$ , $12$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

-2

and

4

can never be equal, which leaves five pairs that involve

m

.

m-4=-2,\quad 3m+2=-2,\quad m-4=4,\quad 3m+2=4,\quad m-4=3m+2

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=2,\quad m=-1\frac{1}{3},\quad m=8,\quad m=\frac{2}{3},\quad m=-3

For example

3m+2=-2

gives

3m=-4

, so

m=-1\frac{1}{3}

, and

m-4=3m+2

gives

m=-3

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-3

,

-1\frac{1}{3}

,

\frac{2}{3}

,

2

,

8

4 · Reviewdoes it hold up?

Check $m=-3$ : the $x$ entries are $-2$ , $-7$ , $4$ , $-7$ , so $x=-7$ goes to both $6$ and $12$ . Check $m=\frac{2}{3}$ : the entries are $-2$ , $-3\frac{1}{3}$ , $4$ , $4$ , so $x=4$ goes to both $9$ and $12$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $-2$ , $-4$ , $4$ , $2$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Variant 12 hard

The table below shows the value of $y$ matched to each value of $x$ . Find all values of $m$ for which this correspondence is not a function.

$x$	$3$	$2m+4$	$6$	$3m-3$
$y$	$10$	$20$	$30$	$40$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

The four $y$ values $10$ , $20$ , $30$ , $40$ are all different. The correspondence stops being a function exactly when one $x$ value gets two different $y$ values, which happens when two of the $x$ entries are equal. Find every $m$ that makes some pair of $x$ entries equal.

Givens

The $x$ entries are $3$ , $2m+4$ , $6$ , $3m-3$ .
The matching $y$ entries are $10$ , $20$ , $30$ , $40$ .

Unknowns

All values of $m$ that make the correspondence not a function.

Constraints

A function gives exactly one $y$ for each $x$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #16 Count the Complement#13 Convert to Algebra

Instead of asking when it is a function, look at when it fails: two equal $x$ entries with different $y$ entries. List every pair of $x$ entries systematically, and turn each pair into a linear equation in $m$ .

3 · Execute4 carry out the plan

1When does it fail?

#16 Count the Complement 8.F.A.1

Every column has a different

y

. So if two columns share the same

x

, that

x

is matched to two

y

values and the correspondence is not a function. If all four

x

entries are different, each

x

has one

y

and it is a function.

A function may not send one input to two outputs.

2List the pairs

#2 Make a Systematic List 8.F.A.1

Four entries make

6

pairs. The pair

3

and

6

can never be equal, which leaves five pairs that involve

m

.

2m+4=3,\quad 3m-3=3,\quad 2m+4=6,\quad 3m-3=6,\quad 2m+4=3m-3

Going through the pairs in order makes sure none is missed.

3Solve each equation

#13 Convert to Algebra 8.EE.C.7

Each pair gives a one-step or two-step linear equation.

m=-\frac{1}{2},\quad m=2,\quad m=1,\quad m=3,\quad m=7

For example

2m+4=3

gives

2m=-1

, so

m=-\frac{1}{2}

, and

2m+4=3m-3

gives

m=7

.

4Collect the values

#2 Make a Systematic List 8.F.A.1

The five values are all different, and each one makes two

x

entries equal, so each breaks the function.

Any one repeated input is enough to fail.

Answer:

-\frac{1}{2}

,

1

,

2

,

3

,

7

4 · Reviewdoes it hold up?

Check $m=7$ : the $x$ entries are $3$ , $18$ , $6$ , $18$ , so $x=18$ goes to both $20$ and $40$ . Check $m=3$ : the entries are $3$ , $10$ , $6$ , $6$ , so $x=6$ goes to both $30$ and $40$ .

Another way: Treat it as a function of $m$ : for any other $m$ , say $m=0$ , the entries $3$ , $4$ , $6$ , $-3$ are all different, so the table is a function. Only the five values above cause a collision.

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Using 'exactly one output per input' to see that a repeated $x$ breaks the function.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equations that make two $x$ entries equal.

💡Takeaway. To break a function, give one input two outputs, and here that means making two

x

entries equal.

Two equal inputs break a function · 12 practice problems

Generated variants — 12

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Where next?