CAIE A-Level Mathematics AS 1.2.3 One One and Inverse Functions Questions

Practise composite and inverse functions and their domains within the AS mathematics syllabus.

Exam points

form and evaluate composite functions
determine and use inverse functions
state domain and range restrictions

[Maximum number: 3]

Functions f and g are defined by

\begin{aligned} & \mathrm{f}(x)=(x+a)^{2}-a \text { for } x \leqslant-a, \\ & \mathrm{~g}(x)=2 x-1 \text { for } x \in \mathbb{R}, \end{aligned}

where a is a positive constant.

Find an expression for $\mathrm{f}^{-1}(x)$ .

$y=(x+a)^{2}-a$ leading to $(x+a)^{2}=y \pm a$

*M1

x and y may be interchanged initially.

Marking guidance:

Allow ± errors for these method marks.

$x=[ \pm] \sqrt{y \pm a} \pm a$
Alternative method for first 2 marks of Question 8(a)

$x=(y+a)^{2}-a$ leading to $y^{2}+2 a y+a^{2}-a-x[=0]$

*M1

Allow ± errors for this method mark.

$y=\frac{-2 a \pm \sqrt{4 a^{2}-4\left(a^{2}-a-x\right)}}{2}$ $\left[y\right.$ or $\left.f^{-1}(x)=\right]-\sqrt{x+a}-a$
OE
Must choose negative root.

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CAIE A-Level Mathematics AS 1.2.3 One One and Inverse Functions Questions question 1