Functions — Challenge

A function is a machine: one input goes in, exactly one output comes out. The rule is the machine.

What these problems cover

These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

If f(x)=2x+3, what is f(4)?

  1. 8
  2. 9
  3. 11
  4. 14
  5. 24
Worked answer

11

Substitute 4 wherever x appears.

f(4) = 2· 4+3 = 8+3 = 11.

Problem 2

If f(x)=x2-1, what is f(-3)?

  1. -10
  2. -8
  3. 8
  4. 10
  5. 26
Worked answer

8

Substitute -3: f(-3) = (-3)2-1.

Squaring a negative gives a positive: (-3)2 = 9.

So f(-3) = 9-1 = 8. Treating the square as -9 would wrongly give -10.

Problem 3

If f(x)=3x-7, what is f(5)?

  1. 2
  2. 8
  3. 15
  4. 22
  5. -8
Worked answer

8

Substitute 5 wherever x appears: f(5) = 3· 5-7.

Multiplication comes before subtraction, so this is 15-7 = 8.

Subtracting first would give 3(5-7) = -6, a different rule.

Problem 4

If f(x)=x2+4, what is f(10)?

  1. 7
  2. 8
  3. 9
  4. 14
  5. 24
Worked answer

9

Substitute 10: f(10) = 102+4.

5+4 = 9. Adding first and then halving would give 7, a different rule.

Problem 5

If f(x)=x+9, for what value of x is f(x)=15?

  1. 5
  2. 6
  3. 7
  4. 24
  5. 135
Worked answer

6

Set the rule equal to the output: x+9 = 15.

Subtract 9 from both sides: x = 6.

Check: f(6) = 6+9 = 15.

Problem 6

If f(x)=2x and g(x)=x+3, what is f(g(1))?

  1. 2
  2. 5
  3. 8
  4. 11
  5. 6
Worked answer

8

Work from the inside out: g(1) = 1+3 = 4.

Then apply f to that result: f(4) = 2· 4 = 8.

Doing it the other way round gives g(f(1)) = g(2) = 5, which is why 5 is among the options.

Problem 7

If f(x)=4-x, what is f(6)?

  1. -2
  2. 2
  3. 10
  4. -10
  5. 24
Worked answer

-2

Substitute 6: f(6) = 4-6.

4-6 = -2. The order matters here: 6-4 would give 2.

Problem 8

If f(x)=x2+2x, what is f(3)?

  1. 9
  2. 11
  3. 15
  4. 21
  5. 33
Worked answer

15

Substitute 3 into both terms: f(3) = 32+2· 3.

9+6 = 15.

Problem 9

If f(x)=7 for every x, what is f(100)?

  1. 0
  2. 7
  3. 100
  4. 107
  5. 700
Worked answer

7

This is a constant function: the rule ignores the input entirely.

Whatever is put in, the output is 7, so f(100) = 7.

Problem 10

If f(x)=x-12, for what value of x is f(x)=5?

  1. 2
  2. 6
  3. 9
  4. 11
  5. 12
Worked answer

11

Set the rule equal to 5: x-12 = 5.

Multiply both sides by 2: x-1 = 10, so x = 11.

Check: 11-12 = 5.

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