1.02u Functions: definition and vocabulary (domain, range, mapping)

279 questions

Sort by: Default | Easiest first | Hardest first
OCR MEI C3 Q1
4 marks Moderate -0.8
  1. Show algebraically that the function \(\text{f}(x) = \frac{2x}{1-x^2}\) is odd. [2] Fig. 7 shows the curve \(y = \text{f}(x)\) for \(0 \leq x < 4\), together with the asymptote \(x = 1\). \includegraphics{figure_7}
  2. Use the copy of Fig. 7 to complete the curve for \(-4 \leq x \leq 4\). [2]
OCR MEI C3 Q2
4 marks Moderate -0.8
The functions f(x) and g(x) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions fg(x) and gf(x), and state whether these functions are odd, even or neither. [4]
OCR MEI C3 Q4
6 marks Moderate -0.3
Fig. 4 shows the curve \(y = \text{f}(x)\), where \(\text{f}(x) = \sqrt{1 - 9x^2}\), \(-a < x < a\). \includegraphics{figure_4}
  1. Find the value of \(a\). [2]
  2. Write down the range of f(x). [1]
  3. Sketch the curve \(y = \text{f}(\frac{1}{3}x) - 1\). [3]
OCR MEI C3 Q5
4 marks Moderate -0.8
You are given that f(x) and g(x) are odd functions, defined for \(x \in \mathbb{R}\).
  1. Given that s(x) = f(x) + g(x), prove that s(x) is an odd function. [2]
  2. Given that p(x) = f(x)g(x), determine whether p(x) is odd, even or neither. [2]
OCR MEI C3 Q6
5 marks Easy -1.2
  1. State the algebraic condition for the function f(x) to be an even function. What geometrical property does the graph of an even function have? [2]
  2. State whether the following functions are odd, even or neither. (A) \(\text{f}(x) = x^2 - 3\) (B) \(\text{g}(x) = \sin x + \cos x\) (C) \(\text{h}(x) = \frac{1}{x + x^3}\) [3]
OCR MEI C3 Q1
6 marks Moderate -0.3
  1. The function f(x) is defined by $$f(x) = \frac{1-x}{1+x}, x \neq -1.$$ Show that f(f(x)) = x. Hence write down \(f^{-1}(x)\). [3]
  2. The function g(x) is defined for all real x by $$g(x) = \frac{1-x^2}{1+x^2}.$$ Prove that g(x) is even. Interpret this result in terms of the graph of \(y = g(x)\). [3]
OCR MEI C3 Q5
4 marks Moderate -0.3
Write down the conditions for f(x) to be an odd function and for g(x) to be an even function. Hence prove that, if f(x) is odd and g(x) is even, then the composite function gf(x) is even. [4]
OCR H240/03 2020 November Q3
11 marks Moderate -0.8
The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).
  1. Find the range of f. [3]
  2. Give a reason why f has no inverse. [1]
  3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
  4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]
AQA AS Paper 1 2020 June Q2
1 marks Easy -1.8
Given that \(f(x) = 10\) when \(x = 4\), which statement below must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(f(2x) = 5\) when \(x = 4\) \(f(2x) = 10\) when \(x = 2\) \(f(2x) = 10\) when \(x = 8\) \(f(2x) = 20\) when \(x = 4\)
AQA Paper 1 2019 June Q6
8 marks Moderate -0.3
The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$
  1. Find the range of f. [1 mark]
    1. Find \(f^{-1}(x)\) [3 marks]
    2. State the range of \(f^{-1}(x)\) [1 mark]
  3. State the transformation which maps the graph of \(y = f(x)\) onto the graph of \(y = f^{-1}(x)\) [1 mark]
  4. Find the coordinates of the point of intersection of the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) [2 marks]
AQA Paper 1 2024 June Q17
6 marks Moderate -0.8
The function f is defined by $$f(x) = |x| + 1 \quad \text{for } x \in \mathbb{R}$$ The function g is defined by $$g(x) = \ln x$$ where g has its greatest possible domain.
  1. Using set notation, state the range of f [2 marks]
  2. State the domain of g [1 mark]
  3. The composite function h is given by $$h(x) = g f(x) \quad \text{for } x \in \mathbb{R}$$
    1. Write down an expression for \(h(x)\) in terms of \(x\) [1 mark]
    2. Determine if h has an inverse. Fully justify your answer. [2 marks]
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]
AQA Paper 3 2018 June Q6
13 marks Standard +0.8
A function \(f\) is defined by \(f(x) = \frac{x}{\sqrt{2x - 2}}\)
  1. State the maximum possible domain of \(f\). [2 marks]
  2. Use the quotient rule to show that \(f'(x) = \frac{x - 2}{(2x - 2)^{\frac{3}{2}}}\). [3 marks]
  3. Show that the graph of \(y = f(x)\) has exactly one point of inflection. [7 marks]
  4. Write down the values of \(x\) for which the graph of \(y = f(x)\) is convex. [1 mark]
AQA Paper 3 2020 June Q3
1 marks Easy -2.5
Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}
AQA Paper 3 2022 June Q3
1 marks Easy -1.8
The function f is defined by $$f(x) = 2x + 1$$ Solve the equation $$f(x) = f^{-1}(x)$$ Circle your answer. [1 mark] \(x = -1\) \quad\quad \(x = 0\) \quad\quad \(x = 1\) \quad\quad \(x = 2\)
AQA Paper 3 2022 June Q10
13 marks Standard +0.3
The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}
  1. State the value of \(x\) which is not in the domain of f. [1 mark]
  2. Explain how you know that the function f is many-to-one. [2 marks]
    1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
    2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
  4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
  5. Using set notation, state the range of f. [2 marks]
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 Further Unit 4 2023 June Q1
5 marks Standard +0.3
The functions \(f\) and \(g\) have domains \((-1, \infty)\) and \((0, \infty)\) respectively and are defined by $$f(x) = \cosh x, \qquad g(x) = x^2 - 1.$$
  1. State the domain and range of \(fg\). [2]
  2. Solve the equation \(fg(x) = 3\). Give your answer correct to three decimal places. [3]
SPS SPS SM Pure 2021 June Q8
5 marks Moderate -0.3
  1. Given that \(\mathbf{f}(x) = x^2 - 4x + 2\), find \(\mathbf{f}(3 + h)\) Express your answer in the form \(h^2 + bh + c\), where \(b\) and \(c \in \mathbb{Z}\). [2 marks]
  2. The curve with equation \(y = x^2 - 4x + 2\) passes through the point \(P(3, -1)\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\). [3 marks]
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 FM Pure 2023 June Q1
5 marks Easy -1.2
You are given that \(gf(x) = |3x - 1|\) for \(x \in \mathbb{R}\).
  1. Given that \(f(x) = 3x - 1\), express \(g(x)\) in terms of \(x\). [1]
  2. State the range of \(gf(x)\). [1]
  3. Solve the inequality \(|3x - 1| > 1\). [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]
SPS SPS SM Pure 2023 September Q4
8 marks Moderate -0.8
$$f(x) = e^x, x \in \mathbb{R}, x > 0.$$ $$g(x) = 2x^3 + 11, x \in \mathbb{R}.$$
  1. Find and simplify an expression for the composite function \(gf(x)\). [2]
  2. State the domain and range of \(gf(x)\). [2]
  3. Solve the equation $$gf(x) = 27.$$ [3]
The equation \(gf(x) = k\), where \(k\) is a constant, has solutions.
  1. State the range of the possible values of \(k\). [1]
SPS SPS SM 2025 February Q6
9 marks Standard +0.3
For all real values of \(x\), the functions \(f\) and \(g\) are defined by \(f (x) = x^2 + 8ax + 4a^2\) and \(g(x) = 6x - 2a\), where \(a\) is a positive constant.
  1. Find \(fg(x)\). Determine the range of \(fg(x)\) in terms of \(a\). [4]
  2. If \(fg(2) = 144\), find the value of \(a\). [3]
  3. Determine whether the function \(fg\) has an inverse. [2]