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Date	November 2021	Marks available	1	Reference code	21N.1.AHL.TZ0.2
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Sketch	Question number	2	Adapted from	N/A

Question

The function $f$ is defined by $f (x) = \frac{2 x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ .

Write down the equation of

Find the coordinates where the graph of $f$ crosses

the vertical asymptote of the graph of $f$ .

[1]

a.i.

the horizontal asymptote of the graph of $f$ .

[1]

a.ii.

the $x$ -axis.

[1]

b.i.

the $y$ -axis.

[1]

b.ii.

Sketch the graph of $f$ on the axes below.

[1]

The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .

[4]

Markscheme

$x = 3$ A1

[1 mark]

a.i.

$y = - 2$ A1

[1 mark]

a.ii.

$(- 2, 0)$ (accept $x = - 2$ ) A1

[1 mark]

b.i.

$(0, \frac{4}{3})$ (accept $y = \frac{4}{3}$ and $f (0) = \frac{4}{3}$ ) A1

[1 mark]

b.ii.

Note: Award A1 for completely correct shape: two branches in correct quadrants with asymptotic behaviour.

[1 mark]

METHOD 1

$(g (x) =) y = \frac{a x + 4}{3 - x}$

attempt to find $x$ in terms of $y$ (M1)

OR exchange $x$ and $y$ and attempt to find $y$ in terms of $x$

$3 y - x y = a x + 4$ A1

$a x + x y = 3 y - 4$

$x (a + y) = 3 y - 4$

$x = \frac{3 y - 4}{y + a}$

$g^{- 1} (x) = \frac{3 x - 4}{x + a}$ A1

Note: Condone use of $y =$

$g (x) \equiv g^{- 1} (x)$

$\frac{a x + 4}{3 - x} \equiv \frac{3 x - 4}{x + a}$

$\Rightarrow a = - 3$ A1

METHOD 2

$g (x) = \frac{a x + 4}{3 - x}$

attempt to find an expression for $g (g (x))$ and equate to $x$ (M1)

$g g (x) = \frac{a (\frac{a x + 4}{3 - x}) + 4}{3 - (\frac{a x + 4}{3 - x})} = x$ A1

$\frac{a (a x + 4) + 4 (3 - x)}{(9 - 3 x) - (a x + 4)} = x$

$\frac{a (a x + 4) + 4 (3 - x)}{5 - (3 + a) x} = x$

$a (a x + 4) + 4 (3 - x) = x (5 - (3 + a) x)$ A1

equating coefficients of $x^{2}$ (or similar)

$a = - 3$ A1

[4 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection

Show 83 related questions

22M.1.SL.TZ2.1a:
Find $g (0)$ .
22M.1.SL.TZ2.1c:
Find the value of $x$ such that $f (x) = 0$ .
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
21N.1.AHL.TZ0.2a.i:
the vertical asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2a.ii:
the horizontal asymptote of the graph of $f$ .
18M.1.SL.TZ1.T_15b:
Write down the x-coordinate of P and the x-coordinate of Q.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
19M.2.AHL.TZ1.H_4a:
Write down the range of $f$ .
17M.1.AHL.TZ2.H_2a:
Write down the range of $f$ .
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.1.SL.TZ1.T_12b.ii:
Write down the number of solutions to $f(x) = - 6$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
19N.1.SL.TZ0.S_10c:
The line ${L_2}$ is tangent to the graph of $g$ at ${\text{A}}$ and has equation $y = \left( {{\text{ln}}\,p} \right)x + q + 1$ .

The line ${L_2}$ passes through the point $\left( { - 2{\text{, }} - 2} \right)$ .

The gradient of the normal to $g$ at ${\text{A}}$ is $\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}$ .

Find the equation of ${L_1}$ in terms of $x$ .
18M.1.SL.TZ1.T_15c:
Write down the values of x for which $f\left( x \right) > g\left( x \right)$ .
21N.1.AHL.TZ0.2b.ii:
the $y$ -axis.
19M.2.SL.TZ1.T_4a:
Find the value of $f\left( x \right)$ when $x = \frac{1}{2}$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
19N.1.SL.TZ0.S_10a:
Write down the coordinates of ${\text{B}}$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
19N.2.AHL.TZ0.H_3b:
Given that ${f^{ - 1}}\left( a \right) = 3$ , determine the value of $a$ .
17M.1.AHL.TZ2.H_2b:
Find an expression for ${f^{ - 1}}(x)$ .
EXN.1.SL.TZ0.8c:
Find an expression for $f^{- 1} (x)$ , stating its domain.
22M.1.AHL.TZ2.6b:
The range of $f$ is $a \leq y \leq b$ , where $a, b \in ℝ$ .

Find the value of $a$ and the value of $b$ .
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
22M.2.AHL.TZ1.10b.ii:
State the domain and range of $f^{- 1}$ .
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
19N.2.AHL.TZ0.H_3c:
Given that $g\left( x \right) = 2f\left( {x - 1} \right)$ , find the domain and range of $g$ .
20N.1.AHL.TZ0.H_10a.i:
Write down an expression for $f' (x)$ .
20N.1.AHL.TZ0.H_10b.i:
Show that $g^{- 1}$ exists.
20N.1.AHL.TZ0.H_12b:
State the equation of the horizontal asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12d:
Sketch the graph of $y = f (x)$ , stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.
20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
19N.2.AHL.TZ0.H_3a:
Find the value of $\left( {f \circ f} \right)\left( 1 \right)$ .
18M.1.SL.TZ1.T_15a:
Find the range of f.
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
21M.2.SL.TZ1.9e:
The line $L$ is tangent to the graphs of both $f$ and the inverse function $f^{- 1}$ .

Find the shaded area enclosed by the graphs of $f$ and $f^{- 1}$ and the line $L$ .
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
20N.1.AHL.TZ0.H_10b.iii:
Hence find $g^{- 1} (x)$ .
20N.1.AHL.TZ0.H_10a.ii:
Hence, given that $f^{- 1}$ does not exist, show that $b^{2} - 3 a c > 0$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.1.SL.TZ1.T_12b.i:
Draw the line $y = - 6$ on the axes.
19N.1.SL.TZ0.S_10b:
Given that $f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}$ , find the equation of ${L_1}$ in terms of $x$ , $p$ and $q$ .
EXN.1.SL.TZ0.8b:
State the range of $f$ .
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
21M.2.SL.TZ2.5a:
Find the range of $f$ .
17M.2.SL.TZ2.S_3c:
Write down the domain of $g$ .
21N.1.AHL.TZ0.2d:
The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.
16N.1.SL.TZ0.T_9b:
Find the value of $b$ .
16N.1.SL.TZ0.T_9c:
Write down the range of $f(x)$ .
17M.1.SL.TZ1.T_12c:
Find the range of values of $k$ for which $f(x) = k$ has no solution.
20N.1.AHL.TZ0.H_10b.ii:
$g (x)$ can be written in the form $p (x - 2)^{3} + q$ , where $p, q \in ℝ$ .

Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10d:
Sketch the graphs of $y = g (x)$ and $y = g^{- 1} (x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.
20N.1.AHL.TZ0.H_12a:
State the equation of the vertical asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12c:
Use an algebraic method to determine whether $f$ is a self-inverse function.
20N.1.AHL.TZ0.H_12e:
The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
19M.2.AHL.TZ1.H_4b:
Find $f'\left( x \right)$ , stating its domain.
21M.2.AHL.TZ2.12e:
State the domain of $g^{- 1}$ .
20N.1.AHL.TZ0.H_10c:
State each of the transformations in the order in which they are applied.
17N.1.SL.TZ0.S_3c:
On the grid, sketch the graph of ${f^{ - 1}}$ .
EXN.1.SL.TZ0.8d:
Find the coordinates of the point(s) where the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ intersect.
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
17M.1.AHL.TZ2.H_2c:
Write down the domain and range of ${f^{ - 1}}$ .
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
16N.1.SL.TZ0.T_9a:
Write down the equation of the axis of symmetry for this graph.
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.2.SL.TZ1.T_4c:
Find the coordinates of the bridges relative to the centre of Orangeton.
20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of $34.50$ represents.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.
20N.1.SL.TZ0.T_2c:
Kaelani has $450 PGK$ .

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
19M.2.SL.TZ1.T_4b.i:
Find the function, $P\left( x \right)$ , that would define this footpath on the map.

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