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IGCSE Additional Math1.6. Find inverse functionsTopic Practice

1.6. Find inverse functions

CAIE IGCSE Additional Math 1.6. Find inverse functions question practice helps you revise this syllabus point with the course map in view. Use this page to focus on one topic, check the style of questions available, and connect each attempt back to the knowledge area it is testing.

EduNinja keeps Additional Math practice aligned to CAIE, so you can move from topic review into exam-style question bank work without losing the syllabus structure. Start with a small set, mark the weak steps, then return to nearby topic links when a definition, graph, calculation, or explanation needs repair.

Question 2(d)

[Maximum number: 5]

The function f is defined by f(x)=1−4x−x2\mathrm{f}(x)=1-4 x-x^{2} for all real values of x.

Using your value of k, find g−1(x)\mathrm{g}^{-1}(x), stating its domain and range.

The function g is defined by g(x)=1−4x−x2\mathrm{g}(x)=1-4 x-x^{2} for x⩾kx \geqslant k, where k is a constant.

Question 2(b)(i)

A function g is defined by g(x)=ex−2\mathrm{g}(x)=\mathrm{e}^{\sqrt{x-2}} for x⩾2x \geqslant 2.

Find an expression for g−1(x)\mathrm{g}^{-1}(x).

Question 5(d)

[Maximum number: 3]

Using this value of p, find an expression for f−1\mathrm{f}^{-1}.

Question 6(b)(ii)

[Maximum number: 2]

It is given that h(x)=a+bx2\mathrm{h}(x)=a+\frac{b}{x^{2}}, where a and b are constants.

Given that h(1)=4 and h′(1)=16\mathrm{h}^{\prime}(1)=16, find the values of a and b.

Question 6(b)(iii)

[Maximum number: 3]

It is given that h(x)=(x−1)2+3\mathrm{h}(\mathrm{x})=(\mathrm{x}-1)^{2}+3 for x⩾a\mathrm{x} \geqslant \mathrm{a}. The value of a is as small as possible such that h−1\mathrm{h}^{-1} exists.

Find h−1(x)\mathrm{h}^{-1}(\mathrm{x}) and state its domain.

Question 9(b)

[Maximum number: 4]

The functions f and g are defined as follows, for all real values of x.

f(x)=2x2−1 g(x)=ex+1\begin{aligned} & \mathrm{f}(x)=2 x^{2}-1 \\ & \mathrm{~g}(x)=\mathrm{e}^{x}+1 \end{aligned}

For each of the functions f and g , either explain why the inverse function does not exist or find the inverse function, stating its domain.

Question 7(a)

[Maximum number: 3]

f(x)=3+(4x−2)5f(x)=\sqrt{3+(4 x-2)^{5}} where x>1.
Find an expression for f′(x)\mathrm{f}^{\prime}(\mathrm{x}), giving your answer as a simplified algebraic fraction.

Question 8(b)(i)

[Maximum number: 2]

The functions g and h are defined by

g(x)=8x3+33 for x⩾1 h(x)=e4x for x⩾k\begin{array}{ll} \mathrm{g}(\mathrm{x})=\sqrt[3]{8 \mathrm{x}^{3}+3} & \text { for } \mathrm{x} \geqslant 1 \\ \mathrm{~h}(\mathrm{x})=\mathrm{e}^{4 \mathrm{x}} & \text { for } x \geqslant k \end{array}

Find an expression for g−1(x)\mathrm{g}^{-1}(\mathrm{x}).

Question 9(b)(ii)

[Maximum number: 4]

The function g is defined, for x⩾1\mathrm{x} \geqslant 1, by g(x)=x2+2x−1\mathrm{g}(\mathrm{x})=\sqrt{\mathrm{x}^{2}+2 \mathrm{x}-1}.

Show that g−1(x)=−1+px2+q\mathrm{g}^{-1}(\mathrm{x})=-1+\sqrt{p x^{2}+q}, where p and q are integers.

Question 9(c)

[Maximum number: 4]

The functions f and g are defined by

f(x)=3x24x−1 for x<0 g(x)=1x2 for x<0\begin{array}{ll} \mathrm{f}(x)=\frac{3 x^{2}}{4 x-1} & \text { for } x<0 \\ \mathrm{~g}(x)=\frac{1}{x^{2}} & \text { for } x<0 \end{array}

Show that f−1(x)\mathrm{f}^{-1}(x) can be written as px−x(qx+r)3\frac{p x-\sqrt{x(q x+r)}}{3} where p, q and r are integers.

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