Depth · one concept from first principles · step 2 of 4
Two equal inputs break a function
The table below shows the value of matched to each value of . Find all values of for which this correspondence is not a function.
FunctionsExpressions
Your answer
How to solve
Strategy Make a Systematic List — 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.
1STEP 1
When does it fail?
All the y values differ, so the table fails to be a function exactly when two x entries are equal.
A function may not send one input to two outputs.
8.F.A.1Change Focus Count The ComplementThe table fails to be a function exactly when two columns have the same .
Why?
The four values are all different, so an shared by two columns would be matched to two different outputs.
Why?
A function sends every input to exactly one output, so one input with two outputs is enough to break it.
🧱A function gives one output per inputCalling a rule a function means every allowed input is sent to exactly one output, never none and never two, so one input with two outputs is enough to break it.
2STEP 2
List the pairs
Four entries make 6 pairs. The pair 2 and 5 can never be equal, which leaves five pairs that involve m.
m-1=2, 2m+1=2, m-1=5, 2m+1=5, m-1=2m+1
Going through the pairs in order makes sure none is missed.
8.F.A.1Make A Systematic List3STEP 3
Solve each equation
Each pair gives a one-step or two-step linear equation.
m=3, m=1/2, m=6, m=2, m=-2
For example 2m+1=2 gives 2m=1, so m=1/2, and m-1=2m+1 gives m=-2.
8.EE.C.7Convert To Algebra4STEP 4
Collect the values
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.
8.F.A.1Make A Systematic ListAnswer
-2, 1/2, 2, 3, 6
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.
Takeaway
To break a function, give one input two outputs, and here that means making two x entries equal.
- When does it fail?
- List the pairs
- Solve each equation
- Collect the values
Where next?
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