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Maths · Functions, Sequences and Rates of Change

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Functions and Function Notation

Using f(x) notation, composite functions such as fg(x), and inverse functions.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Use function notation, such as \(f(x) = 3x - 5\), to evaluate a function and to solve \(f(x) = k\).
  2. 2Write and evaluate composite functions such as \(fg(x)\) and \(gf(x)\), in the correct order.
  3. 3Find the inverse function \(f^{-1}(x)\) of a function.
  4. 4Use functions in problems, including equations that involve two functions.

Functions as machines

A function is a rule that turns each input into exactly one output. Function notation writes this rule as \(f(x)\), read as "f of x". The brackets do not mean multiply: \(f(4)\) means the output when the input is 4. Functions are a Higher tier topic on every board. A function machine is a good way to picture one, and the same ideas of composite and inverse functions follow from joining machines together or running them backwards.

Evaluating and solving

\(f(x) = 2x^2 - 3\). Work out \(f(-2)\), and find the values of \(x\) for which \(f(x) = 5\).

Show the solutionHide the solution
  1. 1 Substitute \(f(-2) = 2 \times (-2)^2 - 3\).
  2. 2 Square first \((-2)^2 = 4\), so \(f(-2) = 8 - 3 = 5\).
  3. 3 Set up an equation \(2x^2 - 3 = 5\), so \(2x^2 = 8\) and \(x^2 = 4\).
  4. 4 Both roots \(x = 2\) or \(x = -2\).

Answer\(f(-2) = 5\), and \(x = 2\) or \(x = -2\)

Composite functions

A composite function is one function applied after another.

  • Notation

    \(fg(x)\) means \(f(g(x))\): do \(g\) first, then \(f\).

  • The order

    The function nearest to \(x\) is applied first, so \(fg(x)\) and \(gf(x)\) are usually different.

  • Method

    Replace the \(x\) in \(f\) by the whole of \(g(x)\), and then simplify.

  • Numbers

    For \(fg(3)\), work out \(g(3)\) first, then put the answer into \(f\).

A composite function

\(f(x) = 2x - 1\) and \(g(x) = x^2\). Work out \(fg(3)\) and find \(gf(x)\), giving your answer in the form \(ax^2 + bx + c\).

Show the solutionHide the solution
  1. 1 Do g first \(g(3) = 3^2 = 9\).
  2. 2 Then f \(fg(3) = f(9) = 2 \times 9 - 1 = 17\).
  3. 3 For gf(x) Do \(f\) first: \(f(x) = 2x - 1\).
  4. 4 Then g \(gf(x) = (2x - 1)^2 = 4x^2 - 4x + 1\).

Answer\(fg(3) = 17\) and \(gf(x) = 4x^2 - 4x + 1\)

Inverse functions

An inverse function reverses a function, and takes the output back to the input.

  • Notation

    \(f^{-1}(x)\) is the inverse of \(f(x)\). The \(-1\) is not a power.

  • Finding it

    Write \(y = f(x)\), make \(x\) the subject, and then swap \(y\) back to \(x\).

  • Check

    \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

  • Machines

    Reverse the order of the operations and use the opposite operation of each.

An inverse function

\(f(x) = \dfrac{x + 4}{3}\). Find \(f^{-1}(x)\), and solve \(f^{-1}(x) = f(x)\).

Show the solutionHide the solution
  1. 1 Write y \(y = \dfrac{x + 4}{3}\).
  2. 2 Rearrange \(3y = x + 4\), so \(x = 3y - 4\).
  3. 3 Swap \(f^{-1}(x) = 3x - 4\).
  4. 4 Solve \(3x - 4 = \dfrac{x + 4}{3}\), so \(9x - 12 = x + 4\), \(8x = 16\) and \(x = 2\).

Answer\(f^{-1}(x) = 3x - 4\) and \(x = 2\)

Test yourself

  1. 1

    What does \(fg(x)\) mean?

    Show answerHide answer

    \(f(g(x))\): do \(g\) first, then \(f\).

  2. 2

    What does \(f^{-1}(x)\) do?

    Show answerHide answer

    It reverses \(f\), taking an output back to the input.

  3. 3

    How do you find an inverse function?

    Show answerHide answer

    Write \(y = f(x)\), make \(x\) the subject, then swap \(y\) for \(x\).

  4. 4

    Are \(fg(x)\) and \(gf(x)\) usually equal?

    Show answerHide answer

    No, the order changes the answer.

  5. 5

    What is \(f(f^{-1}(x))\)?

    Show answerHide answer

    \(x\).

Exam technique: functions

Most errors are about order, so slow down at that point.

  • Write the order

    Underline which function goes first before substituting.

  • Brackets

    Put the whole of the inner function in brackets when you substitute, such as \((2x - 1)^2\).

  • Inverse

    Always check by trying a number.

  • Two functions

    An equation such as \(fg(x) = gf(x)\) needs both sides worked out and then solved.

Summary and exam focus

  • \(f(x)\) is the output from input \(x\), so \(f(a)\) means replace \(x\) by \(a\).
  • \(fg(x) = f(g(x))\), with \(g\) applied first.
  • The inverse \(f^{-1}(x)\) reverses \(f\): make \(x\) the subject, then swap letters.
  • Functions are Higher tier on every board.

Exam focus

\(f(x) = 3x + 2\) and \(g(x) = x^2\). Work out \(fg(x)\) and \(gf(x)\), and find \(f^{-1}(x)\). (5 marks) (5 marks)

\(fg(x) = 3x^2 + 2\) and \(gf(x) = (3x + 2)^2\), because the order is different. Then \(y = 3x + 2\) gives \(x = \dfrac{y - 2}{3}\), so \(f^{-1}(x) = \dfrac{x - 2}{3}\).

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Function
A rule that gives one output for each input.
Function notation
Writing a function as \(f(x)\).
Input
The value put into a function.
Output
The value that comes out of a function.
Composite function
One function applied after another.
Inverse function
A function that reverses another function.
Subject
The letter on its own on one side of an equation.
Substitute
Replace a letter by a number or expression.
Domain
The set of allowed input values.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Core

The diagram shows a function machine for \(f\). (a) Write down an expression for \(f(x)\). [1 mark] (b) Work out \(f(-2)\). [1 mark] (c) Solve \(f(x) = 15\). [1 mark]

A function machine with two operations, used to work out the function f of x.

Mark scheme — 3 marks available

  • (a) \(2x + 7\) — B1
  • (b) \(3\) — B1
  • (c) \(4\) — B1

Model answer

(a) \(f(x) = 2x + 7\). (b) \(f(-2) = 2 \times (-2) + 7 = 3\). (c) \(2x + 7 = 15\), so \(x = 4\).

2. Exam question Work out 4 marks Core

\(f(x) = 3x + 2\) and \(g(x) = x - 4\) (a) Work out \(fg(6)\). [2 marks] (b) Show that \(fg(x) = 3x - 10\). [2 marks]

Mark scheme — 4 marks available

  • (a) \(g(6) = 2\) — M1
  • (a) \(8\) — A1
  • (b) \(3(x - 4) + 2\) — M1
  • (b) \(3x - 12 + 2 = 3x - 10\), with the working shown — A1

Model answer

(a) \(g(6) = 2\), then \(f(2) = 3 \times 2 + 2 = 8\). (b) \(fg(x) = f(x - 4) = 3(x - 4) + 2 = 3x - 12 + 2 = 3x - 10\).

3. Exam question Find 3 marks Core

\(f(x) = 5x - 2\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Work out \(f^{-1}(8)\). [1 mark]

Mark scheme — 3 marks available

  • (a) \(5x = y + 2\) or equivalent — M1
  • (a) \(f^{-1}(x) = \dfrac{x + 2}{5}\) — A1
  • (b) \(2\) — B1

Model answer

(a) \(y = 5x - 2\), so \(x = \dfrac{y + 2}{5}\). So \(f^{-1}(x) = \dfrac{x + 2}{5}\). (b) \(f^{-1}(8) = \dfrac{10}{5} = 2\).

4. Exam question Solve 4 marks Core

\(f(x) = 3x - 1\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Solve \(f^{-1}(x) = f(x)\). [2 marks]

Mark scheme — 4 marks available

  • (a) \(x = \dfrac{y + 1}{3}\) or equivalent — M1
  • (a) \(f^{-1}(x) = \dfrac{x + 1}{3}\) — A1
  • (b) \(\dfrac{x + 1}{3} = 3x - 1\) — M1
  • (b) \(\dfrac{1}{2}\) — A1

Model answer

(a) \(f^{-1}(x) = \dfrac{x + 1}{3}\). (b) \(\dfrac{x + 1}{3} = 3x - 1\), so \(x + 1 = 9x - 3\), \(4 = 8x\) and \(x = \dfrac{1}{2}\).

5. Exam question Find 4 marks Stretch

\(f(x) = \dfrac{x + 1}{x - 2}\) where \(x \ne 2\). Find \(f^{-1}(x)\). [4 marks]

Mark scheme — 4 marks available

  • \(y(x - 2) = x + 1\) — M1
  • \(xy - 2y = x + 1\) — M1
  • \(x(y - 1) = 2y + 1\) — M1
  • \(f^{-1}(x) = \dfrac{2x + 1}{x - 1}\) — A1

Model answer

\(y = \dfrac{x + 1}{x - 2}\), so \(y(x - 2) = x + 1\) and \(xy - 2y = x + 1\). Then \(xy - x = 2y + 1\), so \(x(y - 1) = 2y + 1\) and \(x = \dfrac{2y + 1}{y - 1}\). So \(f^{-1}(x) = \dfrac{2x + 1}{x - 1}\).

6. Exam question Work out 4 marks Stretch

\(f(x) = 2x + 1\) and \(g(x) = ax - 3\), where \(a\) is a constant. \(fg(x) = gf(x)\). Work out the value of \(a\). [4 marks]

Mark scheme — 4 marks available

  • \(fg(x) = 2ax - 5\) — M1
  • \(gf(x) = 2ax + a - 3\) — M1
  • \(-5 = a - 3\) — M1
  • \(-2\) — A1

Model answer

\(fg(x) = 2(ax - 3) + 1 = 2ax - 5\) and \(gf(x) = a(2x + 1) - 3 = 2ax + a - 3\). So \(-5 = a - 3\), which gives \(a = -2\).

7. Multiple choice 1 mark Easier

\(f(x) = 3x - 5\). What is \(f(4)\)?

  1. A 12
  2. B 7 Correct
  3. C 2
  4. D -5

Why: \(3 \times 4 - 5 = 7\).

8. Multiple choice 1 mark Easier

\(f(x) = 2x + 1\). What is \(f(-3)\)?

  1. A \(-5\) Correct
  2. B 7
  3. C \(-6\)
  4. D 5

Why: \(2 \times (-3) + 1 = -5\).

9. Multiple choice 1 mark Core

\(f(x) = x^2 + 1\) and \(g(x) = 2x\). What is \(fg(2)\)?

  1. A 10
  2. B 8
  3. C 5
  4. D 17 Correct

Why: \(g(2) = 4\), then \(f(4) = 16 + 1 = 17\).

10. Multiple choice 1 mark Core

\(f(x) = x + 3\) and \(g(x) = x^2\). What is \(gf(x)\)?

  1. A \(x^2 + 3\)
  2. B \(x^2 + 9\)
  3. C \((x + 3)^2\) Correct
  4. D \(x + 9\)

Why: \(gf(x) = g(f(x)) = (x + 3)^2\).

11. Multiple choice 1 mark Core

\(f(x) = 2x + 1\). What is \(ff(x)\)?

  1. A \(4x + 1\)
  2. B \(4x + 3\) Correct
  3. C \(4x^2 + 1\)
  4. D \(2x + 2\)

Why: \(ff(x) = 2(2x + 1) + 1 = 4x + 3\).

12. Multiple choice 1 mark Easier

What is the inverse of \(f(x) = x + 7\)?

  1. A \(f^{-1}(x) = x - 7\) Correct
  2. B \(f^{-1}(x) = \dfrac{1}{x + 7}\)
  3. C \(f^{-1}(x) = 7 - x\)
  4. D \(f^{-1}(x) = x + 7\)

Why: The inverse reverses the rule, so you subtract 7.

13. Multiple choice 1 mark Core

What is the inverse of \(f(x) = 3x - 5\)?

  1. A \(\dfrac{x - 5}{3}\)
  2. B \(\dfrac{x}{3} + 5\)
  3. C \(\dfrac{1}{3x - 5}\)
  4. D \(\dfrac{x + 5}{3}\) Correct

Why: \(y = 3x - 5\) gives \(x = \dfrac{y + 5}{3}\).

14. Multiple choice 1 mark Stretch

\(f(x) = 5x - 4\). Solve \(f^{-1}(x) = f(x)\).

  1. A \(x = 0\)
  2. B \(x = -1\)
  3. C \(x = 1\) Correct
  4. D \(x = 2\)

Why: \(f^{-1}(x) = \dfrac{x + 4}{5}\), so \(\dfrac{x + 4}{5} = 5x - 4\), which gives \(24x = 24\).

15. Multiple choice 1 mark Stretch

\(f(x) = 2x - 1\) and \(g(x) = ax + 3\), and \(fg(x) = gf(x)\). What is \(a\)?

  1. A \(a = 2\)
  2. B \(a = -2\) Correct
  3. C \(a = 8\)
  4. D \(a = -8\)

Why: \(fg(x) = 2(ax + 3) - 1 = 2ax + 5\) and \(gf(x) = a(2x - 1) + 3 = 2ax - a + 3\), so \(5 = 3 - a\) and \(a = -2\).