A-level · Algebra & functions

The modulus functionExam-style questions

The modulus as a distance, and what follows from that. The graph of y = |x − a| and how to sketch y = |f(x)|; modulus equations, with two cases or by squaring; modulus inequalities, one stretch or two; why a solution must be checked when the other side contains x; and counting solutions with a horizontal line across the V.

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37 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The function $f$ is defined by

$f(x) = 4 - |2x - 3|$ , $\quad x \in \mathbb{R}$ .
(a) Sketch the graph of $y = f(x)$ , stating the coordinates of the vertex and of the points where the graph meets the coordinate axes. (2 marks)
(b) Solve the equation $f(x) = 3x - 1$ . (3 marks)

The function $g$ is defined by

$g(x) = 4 - |2x - 3|$ , $\quad x \ge \dfrac{3}{2}$ .
(c) Find $g^{-1}(x)$ and state the domain of $g^{-1}$ . (2 marks)
Level 4 · 7 marksTry it
2
(a) Sketch on a single diagram the graphs of $y = |2x - 7|$ and $y = 4x - 5$ .

(2 marks)
(b) Solve the inequality $|2x - 7| > 4x - 5$ .

(4 marks)
(c) Hence find the largest integer $N$ satisfying the inequality $\left|2\mathrm{e}^{0.05N} - 7\right| > 4\mathrm{e}^{0.05N} - 5$ .

(2 marks)
Level 4 · 8 marksTry it
3
Figure 1

Figure 1 shows a sketch of the graph with equation

$y = |4x - 12| - 5$

The two straight-line parts of the graph meet at the point $P$ .
(a) Find the coordinates of $P$ .

(2 marks)
(b) Solve the equation

$|4x - 12| - 5 = 6x - 11$

(2 marks)

Given that the line with equation $y = kx - 2$ , where $k$ is a constant, crosses the graph in Figure 1 at two distinct points,
(c) find the range of values of $k$ .

(2 marks)
Level 4 · 6 marksTry it
4
Sketch the graph of $y = |x + 2| - 3$ , stating the coordinates of the vertex and of the points where the graph crosses the axes. (4 marks)
Level 4 · 4 marksTry it
5
(a) State the set of values of $x$ for which $x|x| = x^2$ .

(1 mark)
(b) The integer $n$ satisfies $|n - 2| < 5$ .

(i) Find the greatest possible value of $|3n - 8|$ .

(ii) Find the least possible value of $|3n - 8|$ .

(2 marks)
(c) (i) Solve the equation $|4x - 5| = x + 1$ .

(ii) Explain why the equation $|4x - 5| = 4x + 1$ has exactly one solution, and find this solution.

(4 marks)
Level 4 · 7 marksTry it

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