# IB Math Analysis & Approaches Questionbank-Topic: SL 2.5 Composite functions SL Paper 1

### Question

The graph of y = f (x) for -4 ≤ x ≤ 6 is shown in the following diagram. (a)        Write down the value of

(i)       f (2) ;

(ii)      ( f o f )(2) .                                                                                                                                                             

(b)        Let g(x) = $$\frac{1}{2} f (x) +1$$ for -4 ≤ x ≤ 6 . On the axes above, sketch the graph of g .                   

Ans:

(a) (i) f(2) = 6

(ii) (fof)2=− 2 [2 marks]

(b) ## 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 $$f(x) = {x^2}$$ 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 $$f(x) = lo{g_3}\sqrt x$$ , 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 $$f(x) = \cos 2x$$ and $$g(x) = 2{x^2} – 1$$ .

Find $$f\left( {\frac{\pi }{2}} \right)$$ .


a.

Find $$(g \circ f)\left( {\frac{\pi }{2}} \right)$$ .


b.

Given that $$(g \circ f)(x)$$ can be written as $$\cos (kx)$$ , find the value of k, $$k \in \mathbb{Z}$$ .


c.

## Markscheme

$$f\left( {\frac{\pi }{2}} \right) = \cos \pi$$     (A1)

$$= – 1$$     A1     N2

[2 marks]

a.

$$(g \circ f)\left( {\frac{\pi }{2}} \right) = g( – 1)$$ $$( = 2{( – 1)^2} – 1)$$    (A1)

$$= 1$$     A1     N2

[2 marks]

b.

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

evidence of $$2{\cos ^2}\theta – 1 = \cos 2\theta$$ (seen anywhere)     (M1)

$$(g \circ f)(x) = \cos 4x$$

$$k = 4$$     A1     N2

[3 marks]

c.

## Question

Let $$f(x) = {x^2} + 4$$ and $$g(x) = x – 1$$ .

Find $$(f \circ g)(x)$$ .


a.

The vector $$\left( {\begin{array}{*{20}{c}} 3\\ { – 1} \end{array}} \right)$$ translates the graph of $$(f \circ g)$$ to the graph of h .

Find the coordinates of the vertex of the graph of h .


b.

The vector $$\left( {\begin{array}{*{20}{c}} 3\\ { – 1} \end{array}} \right)$$ translates the graph of $$(f \circ g)$$ to the graph of h .

Show that $$h(x) = {x^2} – 8x + 19$$ .


c.

The vector $$\left( {\begin{array}{*{20}{c}} 3\\ { – 1} \end{array}} \right)$$ translates the graph of $$(f \circ g)$$ to the graph of h .

The line $$y = 2x – 6$$ is a tangent to the graph of h at the point P. Find the x-coordinate of P.


d.

## Markscheme

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

$$(f \circ g)(x) = {(x – 1)^2} + 4$$    $$({x^2} – 2x + 5)$$     A1     N2

[2 marks]

a.

METHOD 1

vertex of $$f \circ g$$ at (1, 4)     (A1)

evidence of appropriate approach     (M1)

e.g. adding $$\left( {\begin{array}{*{20}{c}} 3\\ { – 1} \end{array}} \right)$$ to the coordinates of the vertex of $$f \circ g$$

vertex of h at (4, 3)     A1     N3

METHOD 2

attempt to find $$h(x)$$     (M1)

e.g. $${((x – 3) – 1)^2} + 4 – 1$$ , $$h(x) = (f \circ g)(x – 3) – 1$$

$$h(x) = {(x – 4)^2} + 3$$     (A1)

vertex of h at (4, 3)     A1     N3

[3 marks]

b.

evidence of appropriate approach     (M1)

e.g. $${(x – 4)^2} + 3$$ ,$${(x – 3)^2} – 2(x – 3) + 5 – 1$$

simplifying     A1

e.g. $$h(x) = {x^2} – 8x + 16 + 3$$ , $${x^2} – 6x + 9 – 2x + 6 + 4$$

$$h(x) = {x^2} – 8x + 19$$     AG     N0

[2 marks]

c.

METHOD 1

equating functions to find intersection point     (M1)

e.g. $${x^2} – 8x + 19 = 2x – 6$$ , $$y = h(x)$$

$${x^2} – 10x + 25 + 0$$     A1

evidence of appropriate approach to solve     (M1)

appropriate working     A1

e.g. $${(x – 5)^2} = 0$$

$$x = 5$$  $$(p = 5)$$     A1     N3

METHOD 2

attempt to find $$h'(x)$$     (M1)

$$h(x) = 2x – 8$$     A1

recognizing that the gradient of the tangent is the derivative     (M1)

e.g. gradient at $$p = 2$$

$$2x – 8 = 2$$  $$(2x = 10)$$     A1

$$x = 5$$     A1     N3

[5 marks]

d.

## 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 $$f(x) = 2x – 1$$ 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

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 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 $$f(x) = 6x\sqrt {1 – {x^2}}$$, for $$– 1 \leqslant x \leqslant 1$$, and $$g(x) = \cos (x)$$, for $$0 \leqslant x \leqslant \pi$$.

Let $$h(x) = (f \circ g)(x)$$.

Write $$h(x)$$ in the form $$a\sin (bx)$$, where $$a,{\text{ }}b \in \mathbb{Z}$$.


a.

Hence find the range of $$h$$.


b.

## Markscheme

attempt to form composite in any order     (M1)

eg$$\,\,\,\,\,$$$$f\left( {g(x)} \right),{\text{ }}\cos \left( {6x\sqrt {1 – {x^2}} } \right)$$

correct working     (A1)

eg$$\,\,\,\,\,$$$$6\cos x\sqrt {1 – {{\cos }^2}x}$$

correct application of Pythagorean identity (do not accept $${\sin ^2}x + {\cos ^2}x = 1$$)     (A1)

eg$$\,\,\,\,\,$$$${\sin ^2}x = 1 – {\cos ^2}x,{\text{ }}6\cos x\sin x,{\text{ }}6\cos x \left| \sin x\right|$$

valid approach (do not accept $$2\sin x\cos x = \sin 2x$$)     (M1)

eg$$\,\,\,\,\,$$$$3(2\cos x\sin x)$$

$$h(x) = 3\sin 2x$$    A1     N3

[5 marks]

a.

valid approach     (M1)

eg$$\,\,\,\,\,$$amplitude $$= 3$$, sketch with max and min $$y$$-values labelled, $$– 3 < y < 3$$

correct range     A1     N2

eg$$\,\,\,\,\,$$$$– 3 \leqslant y \leqslant 3$$, $$[ – 3,{\text{ }}3]$$ from $$– 3$$ to 3

Note:     Do not award A1 for $$– 3 < y < 3$$ or for “between $$– 3$$ and 3”.

[2 marks]

b.

## 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.

[N/A]

a.

[N/A]

b.

## Question

Let $$f(x) = 1 + {{\text{e}}^{ – x}}$$ and $$g(x) = 2x + b$$, for $$x \in \mathbb{R}$$, where $$b$$ is a constant.

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


a.

Given that $$\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = – 3$$, find the value of $$b$$.


b.

## Markscheme

attempt to form composite     (M1)

eg$$\,\,\,\,\,$$$$g(1 + {{\text{e}}^{ – x}})$$

correct function     A1     N2

eg$$\,\,\,\,\,$$$$(g \circ f)(x) = 2 + b + 2{{\text{e}}^{ – x}},{\text{ }}2(1 + {{\text{e}}^{ – x}}) + b$$

[2 marks]

a.

evidence of $$\mathop {\lim }\limits_{x \to \infty } (2 + b + 2{{\text{e}}^{ – x}}) = 2 + b + \mathop {\lim }\limits_{x \to \infty } (2{{\text{e}}^{ – x}})$$     (M1)

eg$$\,\,\,\,\,$$$$2 + b + 2{{\text{e}}^{ – \infty }}$$, graph with horizontal asymptote when $$x \to \infty$$

Note:     Award M0 if candidate clearly has incorrect limit, such as $$x \to 0,{\text{ }}{{\text{e}}^\infty },{\text{ }}2{{\text{e}}^0}$$.

evidence that $${{\text{e}}^{ – x}} \to 0$$ (seen anywhere)     (A1)

eg$$\,\,\,\,\,$$$$\mathop {\lim }\limits_{x \to \infty } ({{\text{e}}^{ – x}}) = 0,{\text{ }}1 + {{\text{e}}^{ – x}} \to 1,{\text{ }}2(1) + b = – 3,{\text{ }}{{\text{e}}^{{\text{large negative number}}}} \to 0$$, graph of $$y = {{\text{e}}^{ – x}}$$ or

$$y = 2{{\text{e}}^{ – x}}$$ with asymptote $$y = 0$$, graph of composite function with asymptote $$y = – 3$$

correct working     (A1)

eg$$\,\,\,\,\,$$$$2 + b = – 3$$

$$b = – 5$$     A1     N2

[4 marks]

b.

[N/A]

a.

[N/A]

b.

## Question

Consider a function $$f$$. The line L1 with equation $$y = 3x + 1$$ is a tangent to the graph of $$f$$ when $$x = 2$$

Let $$g\left( x \right) = f\left( {{x^2} + 1} \right)$$ and P be the point on the graph of $$g$$ where $$x = 1$$.

Write down $$f’\left( 2 \right)$$.


a.i.

Find $$f\left( 2 \right)$$.


a.ii.

Show that the graph of g has a gradient of 6 at P.


b.

Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q.

Find the y-coordinate of Q.


c.

## Markscheme

recognize that $$f’\left( x \right)$$ is the gradient of the tangent at $$x$$     (M1)

eg   $$f’\left( x \right) = m$$

$$f’\left( 2 \right) = 3$$  (accept m = 3)     A1 N2

[2 marks]

a.i.

recognize that $$f\left( 2 \right) = y\left( 2 \right)$$     (M1)

eg  $$f\left( 2 \right) = 3 \times 2 + 1$$

$$f\left( 2 \right) = 7$$     A1 N2

[2 marks]

a.ii.

recognize that the gradient of the graph of g is $$g’\left( x \right)$$      (M1)

choosing chain rule to find $$g’\left( x \right)$$      (M1)

eg  $$\frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,u = {x^2} + 1,\,\,u’ = 2x$$

$$g’\left( x \right) = f’\left( {{x^2} + 1} \right) \times 2x$$     A2

$$g’\left( 1 \right) = 3 \times 2$$     A1

$$g’\left( 1 \right) = 6$$     AG N0

[5 marks]

b.

at Q, L1L2 (seen anywhere)      (M1)

recognize that the gradient of L2 is g’(1)  (seen anywhere)     (M1)
eg  m = 6

finding g (1)  (seen anywhere)      (A1)
eg  $$g\left( 1 \right) = f\left( 2 \right),\,\,g\left( 1 \right) = 7$$

attempt to substitute gradient and/or coordinates into equation of a straight line      M1
eg  $$y – g\left( 1 \right) = 6\left( {x – 1} \right),\,\,y – 1 = g’\left( 1 \right)\left( {x – 7} \right),\,\,7 = 6\left( 1 \right) + {\text{b}}$$

correct equation for L2

eg  $$y – 7 = 6\left( {x – 1} \right),\,\,y = 6x + 1$$     A1

correct working to find Q       (A1)
eg   same y-intercept, $$3x = 0$$

$$y = 1$$     A1 N2

[7 marks]

c.