Hard GCSE Functions Questions

Challenging, exam-style GCSE Functions questions with worked solutions. Stretch yourself on the hardest inverse functions, rearranging, composite functions, algebraic composites problems.

inverse functionsrearrangingcomposite functionsalgebraic compositesexpanding bracketsorder matters
GCSE Higher34 questionsStep-by-step solutions
Question 1
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.
Show worked solution

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
Question 2
6 markschallenging
Given f(x)=2x+1f(x) = 2x + 1 and g(x)=x−3g(x) = x - 3, find (fg)−1(x)(fg)^{-1}(x) and evaluate (fg)−1(7)(fg)^{-1}(7).
Show worked solution

Worked solution

  1. Form fg(x)

    fg(x)=f(x−3)fg(x) = f(x - 3)

    Do g first, then f.

  2. Substitute into f

    2(x−3)+12(x - 3) + 1

    The input to f is x - 3.

  3. Expand

    2x−6+12x - 6 + 1

    Multiply out the bracket.

  4. Simplify

    2x−52x - 5

    Combine the constants.

  5. So fg(x)

    fg(x)=2x−5fg(x) = 2x - 5

    This is the composite.

  6. Write y=2x−5y = 2x - 5 to invert

    y=2x−5y = 2x - 5

    Let y be the output.

  7. Add 5

    y+5=2xy + 5 = 2x

    Undo the -5.

  8. Divide by 2

    x=y+52x = \frac{y + 5}{2}

    Undo the multiplication.

  9. Write the inverse

    (fg)−1(x)=x+52(fg)^{-1}(x) = \frac{x + 5}{2}

    Swap y for x.

  10. Evaluate at 7

    (fg)−1(7)=122(fg)^{-1}(7) = \frac{12}{2}

    Substitute x=7x = 7.

  11. Simplify

    66

    12 divided by 2 is 6.

  12. Check forwards

    fg(6)=2(6)−5=7fg(6) = 2(6) - 5 = 7

    The composite sends 6 to 7.

  13. Alternative via g−1f−1g^{-1}f^{-1}

    g−1(x)=x+3, f−1(x)=x−12g^{-1}(x) = x + 3,\ f^{-1}(x) = \frac{x - 1}{2}

    Invert each function.

  14. Compose them at 7

    g−1(f−1(7))=g−1(3)=6g^{-1}(f^{-1}(7)) = g^{-1}(3) = 6

    Same answer, reversed order.

  15. State the answer

    (fg)−1(x)=x+52, (fg)−1(7)=6(fg)^{-1}(x) = \frac{x + 5}{2},\ (fg)^{-1}(7) = 6

    So the value is 6.

Answer
(fg)−1(7)=6(fg)^{-1}(7) = 6
Question 3
5 markschallenging
For f(x)=x2f(x) = x^2, which statement about f(x+2)f(x + 2) is correct?
Show worked solution

Worked solution

  1. Read the notation

    f(x+2)f(x + 2)

    The whole input is x + 2.

  2. Substitute into the rule

    f(x+2)=(x+2)2f(x + 2) = (x + 2)^2

    Replace x with x + 2.

  3. Expand

    (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4

    Use (a+b)2(a + b)^2.

  4. Compare with f(x) + 2

    f(x)+2=x2+2f(x) + 2 = x^2 + 2

    Adding 2 to the output is different.

  5. They differ

    x2+4x+4≠x2+2x^2 + 4x + 4 \ne x^2 + 2

    The expressions are not equal.

  6. Check x=1x = 1 for f(x + 2)

    f(3)=9f(3) = 9

    Evaluate the substitution.

  7. Check x=1x = 1 for f(x) + 2

    f(1)+2=3f(1) + 2 = 3

    Different value.

  8. So substituting is not adding

    input change≠output change\text{input change} \ne \text{output change}

    They are distinct operations.

  9. Graph view of f(x + 2)

    shift left by 2\text{shift left by 2}

    Changing the input shifts horizontally.

  10. Graph view of f(x) + 2

    shift up by 2\text{shift up by 2}

    Changing the output shifts vertically.

  11. Reject adding 2 to the output

    that is f(x)+2\text{that is } f(x) + 2

    Not the same as f(x + 2).

  12. Reject doubling

    the 2 is added inside, not multiplied\text{the 2 is added inside, not multiplied}

    No doubling occurs.

  13. Common error

    treating f(x+2) as f(x)+2\text{treating } f(x + 2) \text{ as } f(x) + 2

    A very common slip.

  14. Reflect

    always substitute the whole new input\text{always substitute the whole new input}

    Put x + 2 everywhere x appears.

  15. Choose the statement

    f(x+2)=x2+4x+4f(x + 2) = x^2 + 4x + 4

    So the first option is correct.

Answer
f(x+2)=(x+2)2=x2+4x+4f(x + 2) = (x + 2)^2 = x^2 + 4x + 4
Question 4
6 markschallenging
The function f(x)=x2+1f(x) = x^2 + 1 is defined for x≥0x \ge 0, and g(x)=2x−3g(x) = 2x - 3. Solve fg(x)=17fg(x) = 17.
Show worked solution

Worked solution

  1. Form fg(x)

    fg(x)=f(2x−3)fg(x) = f(2x - 3)

    Do g first, then f.

  2. Apply f

    (2x−3)2+1(2x - 3)^2 + 1

    Square the input, then add 1.

  3. Set equal to 17

    (2x−3)2+1=17(2x - 3)^2 + 1 = 17

    The output is 17.

  4. Subtract 1

    (2x−3)2=16(2x - 3)^2 = 16

    Isolate the square.

  5. Square root both sides

    2x−3=±42x - 3 = \pm 4

    Remember the plus-or-minus.

  6. First case +4

    2x−3=42x - 3 = 4

    Take the positive root.

  7. Solve

    2x=7, x=3.52x = 7,\ x = 3.5

    The first candidate.

  8. Second case -4

    2x−3=−42x - 3 = -4

    Take the negative root.

  9. Solve

    2x=−1, x=−0.52x = -1,\ x = -0.5

    The second candidate.

  10. Recall f needs input ≥\ge 0

    g(x)≥0g(x) \ge 0

    f is only defined for non-negative inputs.

  11. Check x=3.5x = 3.5

    g(3.5)=4≥0 (valid)g(3.5) = 4 \ge 0 \text{ (valid)}

    This input is allowed.

  12. Check x=−0.5x = -0.5

    g(−0.5)=−4<0 (invalid)g(-0.5) = -4 < 0 \text{ (invalid)}

    This input is rejected by the domain.

  13. So only one solution

    x=3.5x = 3.5

    The domain rules out the other.

  14. Common error

    keeping both roots\text{keeping both roots}

    Always check the domain.

  15. State the answer

    x=3.5x = 3.5

    So x=3.5x = 3.5.

Answer
x=3.5x = 3.5
Question 5
5 markschallenging
Given f(x)=2x+1f(x) = 2x + 1 and fg(x)=2x2−3fg(x) = 2x^2 - 3, find g(x)g(x).
Show worked solution

Worked solution

  1. Write the composite relation

    f(g(x))=2x2−3f(g(x)) = 2x^2 - 3

    fg is given as a rule.

  2. Apply f to g(x)

    2 g(x)+1=2x2−32\,g(x) + 1 = 2x^2 - 3

    f doubles its input and adds 1.

  3. Subtract 1

    2 g(x)=2x2−42\,g(x) = 2x^2 - 4

    Undo the +1.

  4. Divide by 2

    g(x)=x2−2g(x) = x^2 - 2

    Undo the doubling.

  5. So g(x)

    g(x)=x2−2g(x) = x^2 - 2

    This is the inner function.

  6. Check: compute fg(x)

    f(x2−2)=2(x2−2)+1f(x^2 - 2) = 2(x^2 - 2) + 1

    Substitute g into f.

  7. Expand

    2x2−4+12x^2 - 4 + 1

    Multiply out.

  8. Simplify

    2x2−32x^2 - 3

    Matches the given rule.

  9. Check with x=1x = 1

    fg(1)=−1, f(g(1))=f(−1)=−1fg(1) = -1,\ f(g(1)) = f(-1) = -1

    Numbers agree.

  10. Interpret

    we reversed f to reveal g\text{we reversed f to reveal g}

    Undoing the outer function isolates g.

  11. Common error

    forgetting to subtract 1 first\text{forgetting to subtract 1 first}

    Undo the operations in the right order.

  12. Order note

    here f is the outer function\text{here f is the outer function}

    f acts on g's output.

  13. General form

    g(x)=fg(x)−12g(x) = \frac{fg(x) - 1}{2}

    Undo f algebraically.

  14. Reflect

    unknown inner functions can be recovered\text{unknown inner functions can be recovered}

    By undoing the known outer function.

  15. State the answer

    g(x)=x2−2g(x) = x^2 - 2

    So g(x)=x2−2g(x) = x^2 - 2.

Answer
g(x)=x2−2g(x) = x^2 - 2

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