CAIE A-Level Mathematics AS 1.2.2 Range and Composition Questions

Practise finding ranges from formulas and graphs, forming fg or gf in the correct order and deciding whether a composite exists from range and domain restrictions.

Exam points

apply translations and stretches to function graphs
derive transformed function equations
interpret graph transformations in context

[Maximum number: 4]

The function f is defined as follows:

f(x)=\sqrt{x}-1 \text { for } x>1 .

The function h is defined by $\mathrm{h}(x)=\frac{1}{x^{2}+2}$ for $x \geqslant 0$ .

Solve the equation $\mathrm{hf}(x)=\mathrm{f}\left(\frac{25}{16}\right)$ . Give your answer in the form $a+b \sqrt{c}$ , where a, b and c are
integers.

$\mathrm{f}\left(\frac{25}{16}\right)=\frac{1}{4}$
SOI

$\frac{1}{(\sqrt{x}-1)^{2}+2}=\frac{1}{4}$
Equating $\frac{1}{(\sqrt{x}-1)^{2}+2}$ , or their 'simplified'
version, to their $\mathrm{f}\left(\frac{25}{16}\right)$ .

$\left[(\sqrt{x}-1)^{2}+2=4 \text { leading to }\right]$

Or
$[x-2 \sqrt{x}+1+2=4 \text { leading to }]$

Or
$[x-1=2 \sqrt{x} \text { leading to }]$
Simplification as far as $x=\ldots$

Marking guidance:

Allow just + in the results because - can be disregarded at this stage.

Can be implied by the final answer.

Note: $x=1 \pm \sqrt{2}$ scores A0.

$3+2 \sqrt{2}$
Must discount the solution $3-2 \sqrt{2}$ .

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CAIE A-Level Mathematics AS 1.2.2 Range and Composition Questions question 1