GCSE Functions Practice Questions

Free GCSE Functions practice questions with full step-by-step worked solutions. Covers function notation, evaluating functions, squaring, negatives. Practise exam-style problems and check your method.

function notationevaluating functionssquaringnegativessolving f(x)=kcubing
GCSE Higher70 questionsStep-by-step solutions
Question 1
1 markeasy
Given f(x)=2x+3f(x) = 2x + 3, work out f(4)f(4).
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Worked solution

  1. Substitute x=4x = 4

    f(4)=2(4)+3f(4) = 2(4) + 3

    Replace every x in the rule with 4.

  2. Work out the arithmetic

    2(4)+3=8+32(4) + 3 = 8 + 3

    Multiply first, then add.

  3. State the value

    f(4)=11f(4) = 11

    So f(4)=11f(4) = 11.

Answer
f(4)=11f(4) = 11
Question 2
1 markeasy
Given h(x)=x2−1h(x) = x^2 - 1, work out h(−3)h(-3).
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Worked solution

  1. Substitute x=−3x = -3

    h(−3)=(−3)2−1h(-3) = (-3)^2 - 1

    Square -3 with brackets.

  2. Work out

    9−1=89 - 1 = 8

    (-3) squared is 9, then subtract 1.

  3. State the value

    h(−3)=8h(-3) = 8

    So h(−3)=8h(-3) = 8.

Answer
h(−3)=8h(-3) = 8
Question 3
2 marksintermediate
Which statement correctly describes what fg(x)fg(x) means?
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Worked solution

  1. Read the notation

    fg(x)

    Composite notation is read from right to left.

  2. Identify the inner function

    g is next to xg \text{ is next to } x

    g is applied first because it is closest to x.

  3. Apply the outer function

    f(g(x))

    f then acts on the output of g.

  4. It is not multiplication

    fg(x)≠f(x)×g(x)fg(x) \ne f(x) \times g(x)

    This is a common mistake.

  5. Compare with gf(x)

    gf(x)=g(f(x))gf(x) = g(f(x))

    The order usually matters.

  6. Choose the statement

    fg(x)=f(g(x))fg(x) = f(g(x))

    So the first option is correct.

Answer
fg(x)=f(g(x))fg(x) = f(g(x))
Question 4
3 markshard
Given f(x)=x−4f(x) = \sqrt{x - 4}, work out f(20)f(20) and state the domain of ff.
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Worked solution

  1. Substitute x=20x = 20

    f(20)=20−4f(20) = \sqrt{20 - 4}

    Replace x with 20.

  2. Work out the inside

    20−4=1620 - 4 = 16

    Do the subtraction first.

  3. Take the root

    16=4\sqrt{16} = 4

    4 squared is 16.

  4. So f(20)

    f(20)=4f(20) = 4

    The output is 4.

  5. Domain condition

    x−4≥0x - 4 \ge 0

    The inside of a square root cannot be negative.

  6. Solve the condition

    x≥4x \ge 4

    Add 4 to both sides.

  7. State the domain

    x≥4x \ge 4

    Only these inputs are allowed.

  8. Check the boundary

    f(4)=0f(4) = 0

    At x=4x = 4 the output is 0.

  9. Reflect

    outputs are ≥0\text{outputs are } \ge 0

    Square roots are never negative.

  10. State the answer

    f(20)=4, domain x≥4f(20) = 4,\ \text{domain } x \ge 4

    So f(20)=4f(20) = 4 with domain x≥4x \ge 4.

Answer
f(20)=4, domain x≥4f(20) = 4,\ \text{domain } x \ge 4
Question 5
5 markschallenging
A function machine multiplies by 3 then adds 4, so f(x)=3x+4f(x) = 3x + 4. The output is 25. Use the inverse function to find the input.
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Worked solution

  1. Write the function

    f(x)=3x+4f(x) = 3x + 4

    Multiply by 3, then add 4.

  2. Write y=f(x)y = f(x)

    y=3x+4y = 3x + 4

    Let y be the output.

  3. Subtract 4

    y−4=3xy - 4 = 3x

    Undo the +4 first.

  4. Divide by 3

    x=y−43x = \frac{y - 4}{3}

    Undo the multiplication.

  5. Write the inverse

    f−1(x)=x−43f^{-1}(x) = \frac{x - 4}{3}

    Swap y for x.

  6. The output is 25

    f(x)=25f(x) = 25

    We know the machine's output.

  7. Use the inverse

    x=f−1(25)x = f^{-1}(25)

    The inverse recovers the input.

  8. Substitute

    25−43\frac{25 - 4}{3}

    Put 25 into the inverse.

  9. Numerator

    25−4=2125 - 4 = 21

    Do the top first.

  10. Divide

    213=7\frac{21}{3} = 7

    21 divided by 3 is 7.

  11. So the input

    x=7x = 7

    The starting number was 7.

  12. Check forwards

    3(7)+4=253(7) + 4 = 25

    The machine gives 25 from 7.

  13. Machine view

    subtract 4, then divide by 3\text{subtract 4, then divide by 3}

    The inverse machine reverses the steps.

  14. Common error

    dividing before subtracting\text{dividing before subtracting}

    Undo the +4 first, in reverse order.

  15. State the answer

    x=7x = 7

    So the input was 7.

Answer
x=7x = 7

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