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Date	November 2018	Marks available	1	Reference code	18N.3.AHL.TZ0.Hsrg_4
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Give a reason and State	Question number	Hsrg_4	Adapted from	N/A

Question

Consider the functions $f$ , $g$ : $\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}$ defined by

$f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {x + y,\,\,x - y} \right)$ and $g\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {xy,\,\,x + y} \right)$ .

Find $\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .

[3]

a.i.

Find $\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .

[2]

a.ii.

State with a reason whether or not $f$ and $g$ commute.

[1]

b.

Find the inverse of $f$ .

[3]

c.

Markscheme

$\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = f\left( {g\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)$ ( $= f\left( {\left( {xy,\,\,x + y} \right)} \right)$ ) (M1)

$= \left( {xy + x + y,\,\,xy - x - y} \right)$ A1A1

[3 marks]

a.i.

$\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = g\left( {f\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)$

$= g\left( {\left( {x + y,\,\,x - y} \right)} \right)$

$= \left( {\left( {x + y} \right)\left( {x - y} \right),\,\,x + y + x - y} \right)$

$= \left( {{x^2} - {y^2}{\text{,}}\,\,2x} \right)$ A1A1

[2 marks]

a.ii.

no because $f \circ g \ne g \circ f$ R1

Note: Accept counter example.

[1 mark]

b.

$f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {a{\text{,}}\,\,b} \right) \Rightarrow \left( {x + y,\,\,x - y} \right) = \left( {a{\text{,}}\,\,b} \right)$ (M1)

$\left\{ {\begin{array}{*{20}{c}} {x = \frac{{a + b}}{2}} \\ {y = \frac{{a - b}}{2}} \end{array}} \right.$ (M1)

${f^{ - 1}}\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {\frac{{x + y}}{2},\,\,\frac{{x - y}}{2}} \right)$ A1

[3 marks]

c.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 2—Functions » AHL 2.7—Composite functions, finding inverse function incl domain restriction

Show 39 related questions

18M.2.AHL.TZ2.H_10c:
Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.
18M.1.AHL.TZ2.H_10b.ii:
Sketch the graph of $y = g\left( x \right)$ . State the equations of any asymptotes and the coordinates of any intercepts with the axes.
18M.2.AHL.TZ2.H_10d.ii:
Show that $\alpha$ + β < −2.
22M.1.AHL.TZ2.10a:
Find an expression for $f^{- 1} (x)$ . You are not required to state a domain.
22M.2.AHL.TZ1.6b:
Find the value of $p$ and the value of $q$ .
18N.1.AHL.TZ0.H_3b:
Write down the least value of $a$ such that $g$ has an inverse.
18N.1.AHL.TZ0.H_3a:
For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.
19M.2.SL.TZ1.S_9b:
Find u.
18N.1.AHL.TZ0.H_3c.i:
For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
19M.2.SL.TZ1.S_9d.iii:
Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x = 8$ and the tangent line to $f$ at $x = 2$ .
18M.2.AHL.TZ2.H_10a.ii:
With reference to your graph, explain why $f$ is a function on the given domain.
16N.2.SL.TZ0.S_1b:
Find $(g \circ f)(x)$ .
18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
18N.3.AHL.TZ0.Hsrg_4c:
Find the inverse of $f$ .
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
18M.1.AHL.TZ2.H_10b.i:
Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
18M.2.AHL.TZ2.H_10a.iii:
Explain why $f$ has no inverse on the given domain.
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
18M.1.SL.TZ1.S_3c:
On the grid above, sketch the graph of f ⁻¹.
17M.2.SL.TZ2.S_6b:
On the following grid, sketch the graph of $(f \circ g)(x)$ , for $0 \leqslant x \leqslant 2.25$ .
18M.1.AHL.TZ2.H_10c:
The function $h$ is defined by $h\left( x \right) = \sqrt x$ , for $x$ ≥ 0.

State the domain and range of $h \circ g$ .
16N.2.SL.TZ0.S_1a:
Find $f(8)$ .
17M.2.SL.TZ2.S_6c:
The equation $(f \circ g)(x) = k$ has exactly two solutions, for $0 \leqslant x \leqslant 2.25$ . Find the possible values of $k$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
17M.2.SL.TZ2.S_6a:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
16N.2.SL.TZ0.S_1c:
Solve $(g \circ f)(x) = 0$ .
18M.1.AHL.TZ2.H_10a:
Find the inverse function ${f^{ - 1}}$ , stating its domain.
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
18M.2.AHL.TZ2.H_10a.iv:
Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .
21M.1.AHL.TZ2.2c:
Write down the range of $f^{- 1} (x)$ .
18N.3.AHL.TZ0.Hsrg_4a.i:
Find $\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .
21M.1.AHL.TZ2.2b:
Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.
21M.1.AHL.TZ2.2a:
Find the range of $f$ .
18M.2.AHL.TZ2.H_10b:
Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .
18N.3.AHL.TZ0.Hsrg_4a.ii:
Find $\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .

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Topic 2—Functions