Find inverse function

A question is this type if and only if it asks to find an expression for f⁻¹(x) given a function f(x), typically requiring algebraic rearrangement to make x the subject.

110 questions · Moderate -0.1

1.02v Inverse and composite functions: graphs and conditions for existence
Sort by: Default | Easiest first | Hardest first
AQA Paper 1 2024 June Q2
1 marks Easy -1.8
The function f is defined by \(f(x) = e^x + 1\) for \(x \in \mathbb{R}\) Find an expression for \(f^{-1}(x)\) Tick \((\checkmark)\) one box. [1 mark] \(f^{-1}(x) = \ln(x - 1)\) \(\square\) \(f^{-1}(x) = \ln(x) - 1\) \(\square\) \(f^{-1}(x) = \frac{1}{e^x + 1}\) \(\square\) \(f^{-1}(x) = \frac{x - 1}{e}\) \(\square\)
AQA Paper 1 Specimen Q10
10 marks Standard +0.3
The function f is defined by $$f(x) = 4 + 3^{-x}, \quad x \in \mathbb{R}$$
  1. Using set notation, state the range of f [2 marks]
  2. The inverse of f is \(f^{-1}\)
    1. Using set notation, state the domain of \(f^{-1}\) [1 mark]
    2. Find an expression for \(f^{-1}(x)\) [3 marks]
  3. The function g is defined by $$g(x) = 5 - \sqrt{x}, \quad (x \in \mathbb{R} : x > 0)$$
    1. Find an expression for \(gf(x)\) [1 mark]
    2. Solve the equation \(gf(x) = 2\), giving your answer in an exact form. [3 marks]
WJEC Unit 3 2018 June Q12
10 marks Moderate -0.8
  1. Given that \(f\) is a function,
    1. state the condition for \(f^{-1}\) to exist,
    2. find \(ff^{-1}(x)\). [2]
  2. The functions \(g\) and \(h\), are given by $$g(x) = x^2 - 1,$$ $$h(x) = e^x + 1.$$
    1. Suggest a domain for \(g\) such that \(g^{-1}\) exists.
    2. Given the domain of \(h\) is \((-\infty, \infty)\), find an expression for \(h^{-1}(x)\) and sketch, using the same axes, the graphs of \(h(x)\) and \(h^{-1}(x)\). Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
    3. Determine an expression for \(gh(x)\) in its simplest form. [8]
WJEC Unit 3 2023 June Q10
8 marks Moderate -0.8
Two real functions are defined as $$f(x) = \frac{8}{x-4} \quad \text{for} \quad (-\infty < x < 4) \cup (4 < x < \infty),$$ $$g(x) = (x-2)^2 \quad \text{for} \quad -\infty < x < \infty.$$
    1. Find an expression for \(fg(x)\). [2]
    2. Determine the values of \(x\) for which \(fg(x)\) does not exist. [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
WJEC Unit 3 2024 June Q10
14 marks Standard +0.3
The function \(f\) has domain \([4, \infty)\) and is defined by $$f(x) = \frac{2(3x + 1)}{x^2 - 2x - 3} + \frac{x}{x + 1}.$$
  1. Show that \(f(x) = \frac{x + 2}{x - 3}\). [4]
  2. Determine the range of \(f(x)\). [2]
  3. Find an expression for \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\). [4]
  4. Find the value of \(x\) when \(f(x) = f^{-1}(x)\). [4]
WJEC Unit 3 Specimen Q3
8 marks Moderate -0.3
  1. Sketch the graph of \(y = x^2 + 6x + 13\), identifying the stationary point. [2]
  2. The function \(f\) is defined by \(f(x) = x^2 + 6x + 13\) with domain \((a,b)\).
    1. Explain why \(f^{-1}\) does not exist when \(a = -10\) and \(b = 10\). [1]
    2. Write down a value of \(a\) and a value of \(b\) for which the inverse of \(f\) does exist and derive an expression for \(f^{-1}(x)\). [5]
SPS SPS SM Pure 2021 June Q4
5 marks Moderate -0.3
The function \(\mathbf{f}\) is defined by $$\mathbf{f}(x) = \frac{3x - 7}{x - 2} \quad x \in \mathbb{R}, x \neq 2$$
  1. Find \(\mathbf{f}^{-1}(7)\) [2]
  2. Show that \(\mathbf{f}(x) = \frac{ax + b}{x - 3}\) where \(a\) and \(b\) are integers to be found. [3]
SPS SPS SM Pure 2021 May Q1
7 marks Moderate -0.8
The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$
  1. Evaluate \(f(169)\). [2]
  2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
  3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]
SPS SPS SM 2022 February Q8
9 marks Moderate -0.8
The diagram shows the graph of \(y = f(x)\), where \(f(x) = 2 - x^2, \quad x \leqslant 0\). \includegraphics{figure_8}
  1. Evaluate \(f(-3)\). [3]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Sketch the graph of \(y = f^{-1}(x)\). Indicate the coordinates of the points where the graph meets the axes. [3]
SPS SPS SM Pure 2023 June Q11
10 marks Standard +0.3
The function \(f\) is defined by $$f(x) = \frac{12x}{3x + 4} \quad x \in \mathbb{R}, x \geq 0$$
  1. Find the range of \(f\). [2]
  2. Find \(f^{-1}\). [3]
  3. Show, for \(x \in \mathbb{R}, x \geq 0\), that $$ff(x) = \frac{9x}{3x + 1}$$ [3]
  4. Show that \(ff(x) = \frac{7}{2}\) has no solutions. [2]