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Date	May Specimen paper	Marks available	3	Reference code	SPM.1.AHL.TZ0.9
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	9	Adapted from	N/A

Question

The function $f$ $f$ is defined by $f\left( x \right) = {{\text{e}}^{2x}} - 6{{\text{e}}^x} + 5{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \leqslant a$ . The graph of $y = f\left( x \right)$ is shown in the following diagram.

Find the largest value of $a$ such that $f$ has an inverse function.

[3]

a.

For this value of $a$ , find an expression for ${f^{ - 1}}\left( x \right)$ , stating its domain.

[5]

b.

Markscheme

attempt to differentiate and set equal to zero M1

$f'\left( x \right) = 2{{\text{e}}^{2x}} - 6{{\text{e}}^x} = 2{{\text{e}}^x}\left( {{{\text{e}}^x} - 3} \right) = 0$ A1

minimum at $x = {\text{ln}}\,3$

$a = {\text{ln}}\,3$ A1

[3 marks]

a.

Note: Interchanging $x$ and $y$ can be done at any stage.

$y = {\left( {{{\text{e}}^x} - 3} \right)^2} - 4$ (M1)

${{\text{e}}^x} - 3 = \pm \sqrt {y + 4}$ A1

as $x \leqslant {\text{ln}}\,3$ , $x = {\text{ln}}\left( {3 - \sqrt {y + 4} } \right)$ R1

so ${f^{ - 1}}\left( x \right) = {\text{ln}}\left( {3 - \sqrt {x + 4} } \right)$ A1

domain of ${f^{ - 1}}$ is $x \in \mathbb{R}$ , $- 4 \leqslant x < 5$ A1

[5 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 2—Functions » AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

Show 22 related questions

22M.1.AHL.TZ2.6a:
Show that $f$ is an odd function.
22M.1.AHL.TZ2.11b.ii:
State the domain of $g^{- 1}$ .
19M.1.AHL.TZ2.H_11b:
State the range of $g$ .
19M.1.AHL.TZ2.H_11c:
Sketch the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.
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.
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.AHL.TZ1.H_4a:
Write down the range of $f$ .
17N.1.AHL.TZ0.H_11c:
Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .
17N.1.AHL.TZ0.H_11b:
By using mathematical induction, prove that

${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .
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.1.AHL.TZ2.H_11d:
Find the coordinates of P.
SPM.1.AHL.TZ0.9b:
For this value of $a$ , find an expression for ${f^{ - 1}}\left( x \right)$ , stating its domain.
22M.1.AHL.TZ2.11b.i:
Show that $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ .
22M.2.AHL.TZ1.6a:
Show that $f$ is an odd function.
22M.2.AHL.TZ1.10b.i:
Show that the inverse function of $f$ is given by $f^{- 1} (x) = \sqrt{x^{2} + 1}$ .
19M.1.AHL.TZ2.H_11a:
Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
21M.2.AHL.TZ2.12a:
Show that $f$ is an even function.
17N.1.AHL.TZ0.H_11a:
Determine whether ${f_n}$ is an odd or even function, justifying your answer.
19M.2.AHL.TZ1.H_4b:
Find $f'\left( x \right)$ , stating its domain.
17N.1.AHL.TZ0.H_11d:
Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .

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Topic 2—Functions