Depth · one concept from first principles · step 2 of 4

Exactly one prime sum makes a function

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $4$ , $5$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $3 \le a \le 9$ .)

FunctionsOperations

Your answer

How to solve

Strategy Make a Systematic List — Split the job: first check the fixed inputs 1 and 2, then test each candidate a from 3 to 9 in a list, crossing out any a that gets no y or two y values.

1STEP 1

What a function needs here

Each x needs exactly one of x+4, x+5 to be prime: no prime or two primes would break the function.

2STEP 2

Check x = 1 and x = 2

x=1: 1+4=5 is prime, 1+5=6 is not, so 1 goes to 4. x=2: only 2+5=7 is prime, so 2 goes to 5.

3STEP 3

Test each a

For a=3 to 9, only a=4 (8, 9) and a=5 (9, 10) give no prime sum; every other a has exactly one.

4STEP 4

Add the allowed values

The allowed values are 3, 6, 7, 8, 9.

3+6+7+8+9=33

Answer

Spot-check a=6: 6+4=10 is not prime and 6+5=11 is, so 6 goes only to 5. Spot-check a=5: 9 and 10 are both composite, so 5 has no partner. The sum 33 is less than 3+4+…+9=42, as it should be after removing 4 and 5.

Takeaway

A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

What a function needs here
Check x = 1 and x = 2
Test each a
Add the allowed values

Where next?

Level upsuggested

Count functions by multiplying output choices

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Run a counting rule backward to every input

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▶ Practice — 12 problems