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IGCSE Math B4.E Composite functionsTopic Practice

4.E Composite functions

Composite functions ‘fg’ will mean ‘do g first then f’

Question 4(d)

[Maximum number: 2]
Figure 1

Figure 1

Information about the function f is shown in Figure 1.
Given that f is the mapping f:x↦px+qf:x\mapsto px+q, where p and q are constants.

Hence find the value of x for which f(x)=ff(x).

Question 6(a)

[Maximum number: 1]

The functions f and g are defined as

f:x↦3x−1g:x↦3x,x≠0\begin{aligned} f &: x \mapsto 3x - 1 \\ g &: x \mapsto \frac{3}{x},\quad x \ne 0 \end{aligned}

The function h is such that
h(x)=62x−3.h(x)=\frac{6}{2x-3}.

Find gf(2).

Question 12(b)

[Maximum number: 2]

The function f is defined for all values of x by f:x↦x2−2\mathrm{f}: x \mapsto x^{2}-2

The function g is given by
g:x↦12x3+4,x≠−43.g:x\mapsto \frac{12}{x^3+4},\qquad x\ne\sqrt[3]{-4}.
Calculate fg(2).

Question 6

[Maximum number: 7]

The functions f, g and h are defined as
f:x↦x+3,g:x↦x2−2x+3,h:x↦6x,x≠0.\begin{aligned} f&:x\mapsto x+3,\\ g&:x\mapsto x^2-2x+3,\\ h&:x\mapsto \frac6x,\quad x\ne0. \end{aligned}

Question 6(a)

(a)

Find

[ 2 ]

Question 6(a)(ii)

(i)

fh(−14)fh\left(-\frac14\right)

[ 2 ]

Question 6(c)

(b)

Find the two values of x for which hgf(x)=2.

[ 5 ]

Question 8(b)

[Maximum number: 3]

The functions f and g are defined as

f(x)=2x−5 for all values of x g(x)=x2 for all x⩾0\begin{aligned} & \mathrm{f}(x)=2 x-5 \text { for all values of } x \\ & \mathrm{~g}(x)=x^{2} \text { for all } x \geqslant 0 \end{aligned}

Find the positive value of x for which gf(x)=36

Question 21(b)

[Maximum number: 4]

The functions f and g are defined for all values of x by
f(x)=3x−5,g(x)=2x2+1.f(x)=3x-5, \qquad g(x)=2x^2+1.

Solve the equation
f(x)=gf(2).

Question 7(c)

[Maximum number: 4]

The functions f, g and h are defined as

f:x↦x2−2x g:x↦1+2xx≠0 h:x↦5x−4x+3x≠−3\begin{aligned} & \mathrm{f}: x \mapsto x^{2}-2 x \\ & \mathrm{~g}: x \mapsto 1+\frac{2}{x} \quad x \neq 0 \\ & \mathrm{~h}: x \mapsto \frac{5 x-4}{x+3} \quad x \neq-3 \end{aligned}

Solve fg(x)=x54−1fg(x)=\frac{x}{54}-1.

Question 8(c)

[Maximum number: 5]

The equation of the straight line L is
y=2-3x.

Find the values of x for which ff(x)+g(x)=0

Show your working clearly and give your answer in the form

a±bc\frac{a \pm \sqrt{b}}{c}

where a, b and c are integers.

Question 8

[Maximum number: 7]

The function f is such that
f:x↦1−1x,x≠0.f:x\mapsto 1-\frac1x,\quad x\ne0.Solutions of ax2+bx+c=0 are x=−b±b2−4ac2a.\text{Solutions of } ax^2+bx+c=0 \text{ are } x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

Question 8(c)

(a)

Show that ff(x)=f−1(x)ff(x)=f^{-1}(x).

[ 6 ]

Question 8(e)

(b)

Evaluate fff(2).

[ 1 ]

Question 10

[Maximum number: 6]

The function g is such that
g:x↦12x−3.g:x\mapsto\frac{12}{x-3}.

Question 10(d)

(a)

Find hg(5).

[ 2 ]

Question 10(f)

(b)

Show that hg(x) can be written as
4(45−6x+x2)(x−3)2.\frac{4(45-6x+x^2)}{(x-3)^2}.

[ 4 ]
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