# IB Math Analysis & Approaches Questionbank-Topic: SL 2.5 Identity function SL Paper 1

## Question

Let $$f(x) = \ln (x + 5) + \ln 2$$ , for $$x > – 5$$ .

Find $${f^{ – 1}}(x)$$ .


a.

Let $$g(x) = {{\rm{e}}^x}$$ .

Find $$(g \circ f)(x)$$ , giving your answer in the form $$ax + b$$ , where $$a,b \in \mathbb{Z}$$ .


b.

## Markscheme

METHOD 1

$$\ln (x + 5) + \ln 2 = \ln (2(x + 5))$$ $$( = \ln (2x + 10))$$     (A1)

interchanging x and y (seen anywhere)     (M1)

e.g. $$x = \ln (2y + 10)$$

evidence of correct manipulation     (A1)

e.g. $${{\rm{e}}^x} = 2y + 10$$

$${f^{ – 1}}(x) = \frac{{{{\rm{e}}^x} – 10}}{2}$$    A1     N2

METHOD 2

$$y = \ln (x + 5) + \ln 2$$

$$y – \ln 2 = ln(x + 5)$$     (A1)

evidence of correct manipulation     (A1)

e.g. $${{\rm{e}}^{y – \ln 2}} = x + 5$$

interchanging x and y (seen anywhere)     (M1)

e.g. $${{\rm{e}}^{x – \ln 2}} = y + 5$$

$${f^{ – 1}}(x) = {{\rm{e}}^{x – \ln 2}} – 5$$     A1     N2

[4 marks]

a.

METHOD 1

evidence of composition in correct order     (M1)

e.g. $$(g \circ f)(x) = g(\ln (x + 5) + \ln 2)$$

$$= {{\rm{e}}^{\ln (2(x + 5))}} = 2(x + 5)$$

$$(g \circ f)(x) = 2x + 10$$     A1A1     N2

METHOD 2

evidence of composition in correct order     (M1)

e.g. $$(g \circ f)(x) = {{\rm{e}}^{\ln (x + 5) + \ln 2}}$$

$$= {{\rm{e}}^{\ln (x + 5)}} \times {{\rm{e}}^{\ln 2}} = (x + 5)2$$

$$(g \circ f)(x) = 2x + 10$$     A1A1     N2

[3 marks]

b.

## Question

Let $$f(x) = {{\rm{e}}^{x + 3}}$$ .

(i)     Show that $${f^{ – 1}}(x) = \ln x – 3$$ .

(ii)    Write down the domain of $${f^{ – 1}}$$ .


a.

Solve the equation $${f^{ – 1}}(x) = \ln \frac{1}{x}$$ .


b.

## Markscheme

(i) interchanging x and y (seen anywhere)     M1

e.g. $$x = {{\rm{e}}^{y + 3}}$$

correct manipulation     A1

e.g. $$\ln x = y + 3$$ , $$\ln y = x + 3$$

$${f^{ – 1}}(x) = \ln x – 3$$     AG     N0

(ii) $$x > 0$$     A1     N1

[3 marks]

a.

collecting like terms; using laws of logs     (A1)(A1)

e.g. $$\ln x – \ln \left( {\frac{1}{x}} \right) = 3$$ , $$\ln x + \ln x = 3$$ , $$\ln \left( {\frac{x}{{\frac{1}{x}}}} \right) = 3$$ , $$\ln {x^2} = 3$$

simplify     (A1)

e.g. $$\ln x = \frac{3}{2}$$ ,  $${x^2} = {{\rm{e}}^3}$$

$$x = {{\rm{e}}^{\frac{3}{2}}}\left( { = \sqrt {{{\rm{e}}^3}} } \right)$$     A1     N2

[4 marks]

b.

## Question

Let $$f(x) = {x^2}$$ and $$g(x) = 2x – 3$$ .

Find $${g^{ – 1}}(x)$$ .


a.

Find $$(f \circ g)(4)$$ .


b.

## Markscheme

for interchanging x and y (may be done later)     (M1)

e.g. $$x = 2y – 3$$

$${g^{ – 1}}(x) = \frac{{x + 3}}{2}$$ (accept $$y = \frac{{x + 3}}{2},\frac{{x + 3}}{2}$$ )     A1     N2

[2 marks]

a.

METHOD 1

$$g(4) = 5$$     (A1)

evidence of composition of functions     (M1)

$$f(5) = 25$$     A1 N3

METHOD 2

$$f \circ g(x) = {(2x – 3)^2}$$     (M1)

$$f \circ g(4) = {(2 \times 4 – 3)^2}$$     (A1)

= 25     A1     N3

[3 marks]

b.

## Question

Let $$f(x) = 2{x^3} + 3$$ and $$g(x) = {{\rm{e}}^{3x}} – 2$$ .

(i)     Find $$g(0)$$ .

(ii)    Find $$(f \circ g)(0)$$ .


a.

Find $${f^{ – 1}}(x)$$ .


b.

## Markscheme

(i) $$g(0) = {{\rm{e}}^0} – 2$$     (A1)

$$= – 1$$     A1     N2

(ii) METHOD 1

e.g. $$(f \circ g)(0) = f( – 1)$$

correct substitution $$f( – 1) = 2{( – 1)^3} + 3$$     (A1)

$$f( – 1) = 1$$     A1     N3

METHOD 2

attempt to find $$(f \circ g)(x)$$     (M1)

e.g. $$(f \circ g)(x) = f({{\rm{e}}^{3x}} – 2)$$ $$= 2{({{\rm{e}}^{3x}} – 2)^3} + 3$$

correct expression for $$(f \circ g)(x)$$     (A1)

e.g. $$2{({{\rm{e}}^{3x}} – 2)^3} + 3$$

$$(f \circ g)(0) = 1$$     A1     N3

[5 marks]

a.

interchanging x and y (seen anywhere)     (M1)

e.g. $$x = 2{y^3} + 3$$

attempt to solve     (M1)

e.g. $${y^3} = \frac{{x – 3}}{2}$$

$${f^{ – 1}}(x) = \sqrt{{\frac{{x – 3}}{2}}}$$     A1     N3

[3 marks]

b.

## Question

Let $$f(x) = k{\log _2}x$$ .

Given that $${f^{ – 1}}(1) = 8$$ , find the value of $$k$$ .


a.

Find $${f^{ – 1}}\left( {\frac{2}{3}} \right)$$ .


b.

## Markscheme

METHOD 1

recognizing that $$f(8) = 1$$     (M1)

e.g. $$1 = k{\log _2}8$$

recognizing that $${\log _2}8 = 3$$     (A1)

e.g. $$1 = 3k$$

$$k = \frac{1}{3}$$     A1     N2

METHOD 2

attempt to find the inverse of $$f(x) = k{\log _2}x$$     (M1)

e.g. $$x = k{\log _2}y$$ , $$y = {2^{\frac{x}{k}}}$$

substituting 1 and 8     (M1)

e.g. $$1 = k{\log _2}8$$ , $${2^{\frac{1}{k}}} = 8$$

$$k = \frac{1}{{{{\log }_2}8}}$$ $$\left( {k = \frac{1}{3}} \right)$$     A1     N2

[3 marks]

a.

METHOD 1

recognizing that $$f(x) = \frac{2}{3}$$     (M1)

e.g. $$\frac{2}{3} = \frac{1}{3}{\log _2}x$$

$${\log _2}x = 2$$     (A1)

$${f^{ – 1}}\left( {\frac{2}{3}} \right) = 4$$ (accept $$x = 4$$)     A2     N3

METHOD 2

attempt to find inverse of $$f(x) = \frac{1}{3}{\log _2}x$$     (M1)

e.g. interchanging x and y , substituting $$k = \frac{1}{3}$$ into $$y = {2^{\frac{x}{k}}}$$

correct inverse     (A1)

e.g. $${f^{ – 1}}(x) = {2^{3x}}$$ , $${2^{3x}}$$

$${f^{ – 1}}\left( {\frac{2}{3}} \right) = 4$$     A2    N3

[4 marks]

b.

## Question

Let $$f(x) = lo{g_3}\sqrt x$$ , for $$x > 0$$ .

Show that $${f^{ – 1}}(x) = {3^{2x}}$$ .


a.

Write down the range of $${f^{ – 1}}$$ .


b.

Let $$g(x) = {\log _3}x$$ , for $$x > 0$$ .

Find the value of $$({f^{ – 1}} \circ g)(2)$$ , giving your answer as an integer.


c.

## Markscheme

interchanging x and y (seen anywhere)     (M1)

e.g. $$x = \log \sqrt y$$ (accept any base)

evidence of correct manipulation     A1

e.g. $$3^x = \sqrt y$$ , $${3^y} = {x^{\frac{1}{2}}}$$ , $$x = \frac{1}{2}{\log _3}y$$ , $$2y = {\log _3}x$$

$${f^{ – 1}}(x) = {3^{2x}}$$     AG     N0

[2 marks]

a.

$$y > 0$$ , $${f^{ – 1}}(x) > 0$$     A1     N1

[1 mark]

b.

METHOD 1

finding $$g(2) = lo{g_3}2$$ (seen anywhere)     A1

attempt to substitute     (M1)

e.g. $$({f^{ – 1}} \circ g)(2) = {3^{2\log {_3}2}}$$

evidence of using log or index rule     (A1)

e.g. $$({f^{ – 1}} \circ g)(2) = {3^{\log {_3}4}}$$ , $${3^{{{\log }_3}2^2}}$$

$$({f^{ – 1}} \circ g)(2) = 4$$     A1     N1

METHOD 2

attempt to form composite (in any order)     (M1)

e.g. $$({f^{ – 1}} \circ g)(x) = {3^{2{{\log }_3}x}}$$

evidence of using log or index rule     (A1)

e.g. $$({f^{ – 1}} \circ g)(x) = {3^{{{\log }_3}{x^2}}}$$ , $${3^{{{\log }_3}{x^{}}}}^2$$

$$({f^{ – 1}} \circ g)(x) = {x^2}$$     A1

$$({f^{ – 1}} \circ g)(2) = 4$$     A1     N1

[4 marks]

c.

## Question

Let $$f(x) = 7 – 2x$$ and $$g(x) = x + 3$$ .

Find $$(g \circ f)(x)$$ .


a.

Write down $${g^{ – 1}}(x)$$ .


b.

Find $$(f \circ {g^{ – 1}})(5)$$ .


c.

## Markscheme

attempt to form composite     (M1)

e.g. $$g(7 – 2x)$$ , $$7 – 2x + 3$$

$$(g \circ f)(x) = 10 – 2x$$     A1     N2

[2 marks]

a.

$${g^{ – 1}}(x) = x – 3$$     A1     N1

[1 mark]

b.

METHOD 1

valid approach     (M1)

e.g. $${g^{ – 1}}(5)$$ , $$2$$ , $$f(5)$$

$$f(2) = 3$$     A1     N2

METHOD 2

attempt to form composite of f and $${g^{ – 1}}$$     (M1)

e.g. $$(f \circ {g^{ – 1}})(x) = 7 – 2(x – 3)$$ , $$13 – 2x$$

$$(f \circ {g^{ – 1}})(5) = 3$$     A1     N2

[2 marks]

c.

## Question

Let $$f(x) = {\log _p}(x + 3)$$ for $$x > – 3$$ . Part of the graph of f is shown below. The graph passes through A(6, 2) , has an x-intercept at (−2, 0) and has an asymptote at $$x = – 3$$ .

Find p .


a.

The graph of f is reflected in the line $$y = x$$ to give the graph of g .

(i)     Write down the y-intercept of the graph of g .

(ii)    Sketch the graph of g , noting clearly any asymptotes and the image of A.


b.

The graph of $$f$$ is reflected in the line $$y = x$$ to give the graph of $$g$$ .

Find $$g(x)$$ .


c.

## Markscheme

evidence of substituting the point A     (M1)

e.g. $$2 = {\log _p}(6 + 3)$$

manipulating logs     A1

e.g. $${p^2} = 9$$

$$p = 3$$     A2     N2

[4 marks]

a.

(i) $$y = – 2$$ (accept $$(0{\text{, }} – 2))$$     A1     N1

(ii) A1A1A1A1     N4

Note: Award A1 for asymptote at $$y = – 3$$ , A1 for an increasing function that is concave up, A1 for a positive x-intercept and a negative y-intercept, A1 for passing through the point $$(2{\text{, }}6)$$ .

[5 marks]

b.

METHOD 1

recognizing that $$g = {f^{ – 1}}$$     (R1)

evidence of valid approach     (M1)

e.g. switching x and y (seen anywhere), solving for x

correct manipulation     (A1)

e.g. $${3^x} = y + 3$$

$$g(x) = {3^x} – 3$$     A1     N3

METHOD 2

recognizing that $$g(x) = {a^x} + b$$     (R1)

identifying vertical translation     (A1)

e.g. graph shifted down 3 units, $$f(x) – 3$$

evidence of valid approach     (M1)

e.g. substituting point to identify the base

$$g(x) = {3^x} – 3$$     A1     N3

[4 marks]

c.

## Question

Let $$f(x) = 2x – 1$$ and  $$g(x) = 3{x^2} + 2$$ .

Find $${f^{ – 1}}(x)$$ .


a.

Find $$(f \circ g)(1)$$ .


b.

## Markscheme

interchanging x and y (seen anywhere)     (M1)

e.g. $$x = 2y – 1$$

correct manipulation     (A1)

e.g. $$x + 1 = 2y$$

$${f^{ – 1}}(x) = \frac{{x + 1}}{2}$$      A1     N2

[3 marks]

a.

METHOD 1

attempt to find or $$g(1)$$ or $$f(1)$$     (M1)

$$g(1) = 5$$     (A1)

$$f(5) = 9$$     A1     N2

[3 marks]

METHOD 2

attempt to form composite (in any order)     (M1)

e.g. $$2(3{x^2} + 2) – 1$$ , $$3{(2x – 1)^2} + 2$$

$$(f \circ g)(1) = 2(3 \times {1^2} + 2) – 1$$ $$( = 6 \times {1^2} + 3)$$     (A1)

$$(f \circ g)(1) = 9$$     A1     N2

[3 marks]

b

## Question

Let $$f(x) = \sqrt {x – 5}$$ , for $$x \ge 5$$ .

Find $${f^{ – 1}}(2)$$ .


a.

Let $$g$$ be a function such that $${g^{ – 1}}$$ exists for all real numbers. Given that $$g(30) = 3$$ , find $$(f \circ {g^{ – 1}})(3)$$  .


b.

## Markscheme

METHOD 1

attempt to set up equation     (M1)

eg   $$2 = \sqrt {y – 5}$$ , $$2 = \sqrt {x – 5}$$

correct working     (A1)

eg   $$4 = y – 5$$ , $$x = {2^2} + 5$$

$${f^{ – 1}}(2) = 9$$     A1     N2

METHOD 2

interchanging $$x$$ and $$y$$ (seen anywhere)     (M1)

eg   $$x = \sqrt {y – 5}$$

correct working     (A1)

eg   $${x^2} = y – 5$$ , $$y = {x^2} + 5$$

$${f^{ – 1}}(2) = 9$$     A1     N2

[3 marks]

a.

recognizing $${g^{ – 1}}(3) = 30$$     (M1)

eg   $$f(30)$$

correct working     (A1)

eg   $$(f \circ {g^{ – 1}})(3) = \sqrt {30 – 5}$$ , $$\sqrt {25}$$

$$(f \circ {g^{ – 1}})(3) = 5$$     A1     N2

Note: Award A0 for multiple values, eg $$\pm 5$$ .

[3 marks]

b.

## Question

Let $$f(x) = 4x – 2$$ and $$g(x) = – 2{x^2} + 8$$ .

Find $${f^{ – 1}}(x)$$ .


a.

Find $$(f \circ g)(1)$$ .


b.

## Markscheme

interchanging $$x$$ and $$y$$ (seen anywhere)     (M1)

eg   $$x = 4y – 2$$

evidence of correct manipulation     (A1)

eg   $$x + 2 = 4y$$

$${f^{ – 1}}(x) = \frac{{x + 2}}{4}$$ (accept $$y = \frac{{x + 2}}{4}$$ , $$\frac{{x + 2}}{4}$$ , $${f^{ – 1}}(x) = \frac{1}{4}x + \frac{1}{2}$$     A1     N2

[3 marks]

a.

METHOD 1

attempt to substitute $$1$$ into $$g(x)$$     (M1)

eg   $$g(1) = – 2 \times {1^2} + 8$$

$$g(1) = 6$$     (A1)

$$f(6) = 22$$     A1     N3

METHOD 2

attempt to form composite function (in any order)     (M1)

eg   $$(f \circ g)(x) = 4( – 2{x^2} + 8) – 2$$ $$( = – 8{x^2} + 30)$$

correct substitution

eg   $$(f \circ g)(1) = 4( – 2 \times {1^2} + 8) – 2$$ , $$– 8 + 30$$

$$f(6) = 22$$     A1     N3

[3 marks]

b.

## Question

The diagram below shows the graph of a function $$f$$ , for $$– 1 \le x \le 2$$ . Write down the value of $$f(2)$$.


a.i.

Write down the value of $${f^{ – 1}}( – 1)$$ .


a.ii.

Sketch the graph of $${f^{ – 1}}$$ on the grid below. b.

## Markscheme

$$f(2) = 3$$     A1     N1

[1 mark]

a.i.

$${f^{ – 1}}( – 1) = 0$$     A2     N2

[2 marks]

a.ii.

EITHER

attempt to draw $$y = x$$ on grid     (M1)

OR

attempt to reverse x and y coordinates     (M1)

eg   writing or plotting at least two of the points

$$( – 2, – 1)$$ , $$( – 1,0)$$ , $$(0,1)$$ , $$(3,2)$$

THEN

correct graph     A2     N3 [3 marks]

b.

## Question

Let $$f(x) = 3x – 2$$ and $$g(x) = \frac{5}{{3x}}$$, for $$x \ne 0$$.

Let $$h(x) = \frac{5}{{x + 2}}$$, for $$x \geqslant 0$$. The graph of h has a horizontal asymptote at $$y = 0$$.

Find $${f^{ – 1}}(x)$$.


a.

Show that $$\left( {g \circ {f^{ – 1}}} \right)(x) = \frac{5}{{x + 2}}$$.


b.

Find the $$y$$-intercept of the graph of $$h$$.


c(i).

Hence, sketch the graph of $$h$$.


c(ii).

For the graph of $${h^{ – 1}}$$, write down the $$x$$-intercept;


d(i).

For the graph of $${h^{ – 1}}$$, write down the equation of the vertical asymptote.


d(ii).

Given that $${h^{ – 1}}(a) = 3$$, find the value of $$a$$.


e.

## Markscheme

interchanging $$x$$ and $$y$$     (M1)

eg     $$x = 3y – 2$$

$${f^{ – 1}}(x) = \frac{{x + 2}}{3}{\text{ }}\left( {{\text{accept }}y = \frac{{x + 2}}{3},{\text{ }}\frac{{x + 2}}{3}} \right)$$     A1     N2

[2 marks]

a.

attempt to form composite (in any order)     (M1)

eg     $$g\left( {\frac{{x + 2}}{3}} \right),{\text{ }}\frac{{\frac{5}{{3x}} + 2}}{3}$$

correct substitution     A1

eg     $$\frac{5}{{3\left( {\frac{{x + 2}}{3}} \right)}}$$

$$\left( {g \circ {f^{ – 1}}} \right)(x) = \frac{5}{{x + 2}}$$     AG     N0

[2 marks]

b.

valid approach     (M1)

eg     $$h(0),{\text{ }}\frac{5}{{0 + 2}}$$

$$y = \frac{5}{2}{\text{ }}\left( {{\text{accept (0, 2.5)}}} \right)$$     A1     N2

[2 marks]

c(i).  A1A2     N3

Notes:     Award A1 for approximately correct shape (reciprocal, decreasing, concave up).

Only if this A1 is awarded, award A2 for all the following approximately correct features: y-intercept at $$(0, 2.5)$$, asymptotic to x-axis, correct domain $$x \geqslant 0$$.

If only two of these features are correct, award A1.

[3 marks]

c(ii).

$$x = \frac{5}{2}{\text{ }}\left( {{\text{accept (2.5, 0)}}} \right)$$     A1     N1

[1 mark]

d(i).

$$x = 0$$   (must be an equation)     A1     N1

[1 mark]

d(ii).

METHOD 1

attempt to substitute $$3$$ into $$h$$ (seen anywhere)     (M1)

eg     $$h(3),{\text{ }}\frac{5}{{3 + 2}}$$

correct equation     (A1)

eg     $$a = \frac{5}{{3 + 2}},{\text{ }}h(3) = a$$

$$a = 1$$     A1     N2

[3 marks]

METHOD 2

attempt to find inverse (may be seen in (d))     (M1)

eg     $$x = \frac{5}{{y + 2}},{\text{ }}{h^{ – 1}} = \frac{5}{x} – 2,{\text{ }}\frac{5}{x} + 2$$

correct equation, $$\frac{5}{x} – 2 = 3$$     (A1)

$$a = 1$$     A1     N2

[3 marks]

e.

## Question

The following diagram shows the graph of $$y = f(x)$$, for $$– 4 \le x \le 5$$. Write down the value of $$f( – 3)$$.


a(i).

Write down the value of  $${f^{ – 1}}(1)$$.


a(ii).

Find the domain of $${f^{ – 1}}$$.


b.

On the grid above, sketch the graph of $${f^{ – 1}}$$.


c.

## Markscheme

$$f( – 3) = – 1$$     A1     N1

[1 mark]

a(i).

$${f^{ – 1}}(1) = 0$$   (accept $$y = 0$$)     A1     N1

[1 mark]

a(ii).

domain of $${f^{ – 1}}$$ is range of $$f$$     (R1)

eg     $${\text{R}}f = {\text{D}}{f^{ – 1}}$$

eg     $$– 3 \leqslant x \leqslant 3,{\text{ }}x \in [ – 3,{\text{ }}3]{\text{ (accept }} – 3 < x < 3,{\text{ }} – 3 \leqslant y \leqslant 3)$$

[2 marks]

b. A1A1     N2

Note: Graph must be approximately correct reflection in $$y = x$$.

Only if the shape is approximately correct, award the following:

A1 for x-intercept at $$1$$, and A1 for endpoints within circles.

[2 marks]

c.

## Question

The following diagram shows the graph of a function $$f$$. Find $${f^{ – 1}}( – 1)$$.


a.

Find $$(f \circ f)( – 1)$$.


b.

On the same diagram, sketch the graph of $$y = f( – x)$$.


c.

## Markscheme

valid approach     (M1)

eg$$\;\;\;$$horizontal line on graph at $$– 1,{\text{ }}f(a) = – 1,{\text{ }}( – 1,5)$$

$${f^{ – 1}}( – 1) = 5$$     A1     N2

[2 marks]

a.

attempt to find $$f( – 1)$$     (M1)

eg$$\;\;\;$$line on graph

$$f( – 1) = 2$$     (A1)

$$(f \circ f)( – 1) = 1$$     A1     N3

[3 marks]

b. A1A1     N2

Note:     The shape must be an approximately correct shape (concave down and increasing). Only if the shape is approximately correct, award the following for points in circles:

A1 for the $$y$$-intercept,

A1 for any two of these points $$( – 5,{\text{ }} – 1),{\text{ }}( – 2,{\text{ }}1),{\text{ }}(1,{\text{ }}2)$$.

[2 marks]

Total [7 marks]

c.

## Question

Let $$f(x) = {(x – 5)^3}$$, for $$x \in \mathbb{R}$$.

Find $${f^{ – 1}}(x)$$.


a.

Let $$g$$ be a function so that $$(f \circ g)(x) = 8{x^6}$$. Find $$g(x)$$.


b.

## Markscheme

interchanging $$x$$ and $$y$$ (seen anywhere)     (M1)

eg$$\;\;\;x = {(y – 5)^3}$$

evidence of correct manipulation     (A1)

eg$$\;\;\;y – 5 = \sqrt{x}$$

$${f^{ – 1}}(x) = \sqrt{x} + 5\;\;\;({\text{accept }}5 + {x^{\frac{1}{3}}},{\text{ }}y = 5 + \sqrt{x})$$     A1     N2

Notes:     If working shown, and they do not interchange $$x$$ and $$y$$, award A1A1M0 for $$\sqrt{y} + 5$$.

If no working shown, award N1 for $$\sqrt{y} + 5$$.

a.

METHOD 1

attempt to form composite (in any order)     (M1)

eg$$\;\;\;g\left( {{{(x – 5)}^3}} \right),{\text{ }}{\left( {g(x) – 5} \right)^3} = 8{x^6},{\text{ }}f(2{x^2} + 5)$$

correct working     (A1)

eg$$\;\;\;g – 5 = 2{x^2},{\text{ }}{\left( {(2{x^2} + 5) – 5} \right)^3}$$

$$g(x) = 2{x^2} + 5$$     A1     N2

METHOD 2

recognising inverse relationship     (M1)

eg$$\;\;\;{f^{ – 1}}(8{x^6}) = g(x),{\text{ }}{f^{ – 1}}(f \circ g)(x) = {f^{ – 1}}(8{x^6})$$

correct working

eg$$\;\;\;g(x) = \sqrt{{(8{x^6})}} + 5$$     (A1)

$$g(x) = 2{x^2} + 5$$     A1     N2

b.

## Question

Let $$f(x) = 8x + 3$$ and $$g(x) = 4x$$, for $$x \in \mathbb{R}$$.

Write down $$g(2)$$.


a.

Find $$(f \circ g)(x)$$.


b.

Find $${f^{ – 1}}(x)$$.


c.

## Markscheme

$$g(2) = 8$$    A1     N1

[1 mark]

a.

attempt to form composite (in any order)     (M1)

eg$$\,\,\,\,\,$$$$f(4x),{\text{ }}4 \times (8x + 3)$$

$$(f \circ g)(x) = 32x + 3$$     A1     N2

[2 marks]

b.

interchanging $$x$$ and $$y$$ (may be seen at any time)     (M1)

eg$$\,\,\,\,\,$$$$x = 8y + 3$$

$${f^{ – 1}}(x) = \frac{{x – 3}}{8}\,\,\,\,\,\left( {{\text{accept }}\frac{{x – 3}}{8},{\text{ }}y = \frac{{x – 3}}{8}} \right)$$     A1     N2

[2 marks]

c.

## Question

Let $$f(x) = 5x$$ and $$g(x) = {x^2} + 1$$, for $$x \in \mathbb{R}$$.

Find $${f^{ – 1}}(x)$$.


a.

Find $$(f \circ g)(7)$$.


b.

## Markscheme

interchanging $$x$$ and $$x$$     (M1)

eg$$\,\,\,\,\,$$$$x = 5y$$

$${f^{ – 1}}\left( x \right) = \frac{x}{5}$$     A1     N2

[2 marks]

a.

METHOD 1

attempt to substitute 7 into $$g(x)$$ or $$f(x)$$     (M1)

eg$$\,\,\,\,\,$$$${7^2} + 1,{\text{ }}5 \times 7$$

$$g(7) = 50$$     (A1)

$$f\left( {50} \right) = 250$$     A1     N2

METHOD 2

attempt to form composite function (in any order)     (M1)

eg$$\,\,\,\,\,$$$$5({x^2} + 1),{\text{ }}{(5x)^2} + 1$$

correct substitution     (A1)

eg$$\,\,\,\,\,$$$$5 \times ({7^2} + 1)$$

$$(f \circ g)(7) = 250$$     A1     N2

[3 marks]

b.

## Question

The following diagram shows the graph of a function $$f$$, with domain $$– 2 \leqslant x \leqslant 4$$. The points $$( – 2,{\text{ }}0)$$ and $$(4,{\text{ }}7)$$ lie on the graph of $$f$$.

Write down the range of $$f$$.


a.

Write down $$f(2)$$;


b.i.

Write down $${f^{ – 1}}(2)$$.


b.ii.

On the grid, sketch the graph of $${f^{ – 1}}$$.


c.

## Markscheme

correct range (do not accept $$0 \leqslant x \leqslant 7$$)     A1     N1

eg$$\,\,\,\,\,$$$$[0,{\text{ }}7],{\text{ }}0 \leqslant y \leqslant 7$$

[1 mark]

a.

$$f(2) = 3$$     A1     N1

[1 mark]

b.i.

$${f^{ – 1}}(2) = 0$$     A1     N1

[1 mark]

b.ii. A1A1A1     N3

Notes:     Award A1 for both end points within circles,

A1 for images of $$(2,{\text{ }}3)$$ and $$(0,{\text{ }}2)$$ within circles,

A1 for approximately correct reflection in $$y = x$$, concave up then concave down shape (do not accept line segments).

[3 marks]

c.

## Question

Consider a function f (x) , for −2 ≤ x ≤ 2 . The following diagram shows the graph of f. Write down the value of f (0).


a.i.

Write down the value of f −1 (1).


a.ii.

Write down the range of f −1.


b.

On the grid above, sketch the graph of f −1.


c.

## Markscheme

$$f\left( 0 \right) = – \frac{1}{2}$$     A1 N1

[1 mark]

a.i.

f −1 (1) = 2     A1 N1

[1 mark]

a.ii.

−2 ≤ y ≤ 2, y∈ [−2, 2]  (accept −2 ≤ x ≤ 2)     A1 N1

[1 mark]

b. A1A1A1A1  N4

Note: Award A1 for evidence of approximately correct reflection in y = x with correct curvature.

(y = x does not need to be explicitly seen)

Only if this mark is awarded, award marks as follows:

A1 for both correct invariant points in circles,

A1 for the three other points in circles,

A1 for correct domain.

[4 marks]

c.