NYJC/2021/JC2/Prelim/P1/Q05

(a) The curve $y= f (x)$ has a horizontal asymptote $y=k$ and cuts the axes at $(a, 0)$ and $(0, b)$, where $a$, $b$ and $k$ are non-zero constants. It is given that $f ^{-1}(x)$ exists. State, if possible, the coordinates of the points where the following curves cut the axes and the equations of their asymptotes.

(i) $y= f (2 x-3)$
(ii) $y= f ^{-1}(x)$

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(b) The function $g$ is given by $g : x \mapsto ln \left(\frac{ e ^x+5}{ e ^x-1}\right), \text {, for } x \in \mathbb{R} , x>0$.
(i) Find $g ^{-1}(x)$ and state its domain.

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The function $h$ is defined by
$h : x \mapsto 1+\sqrt{9-(x-2)^2} \text {, for } x \in \mathbb{R} , -1 \leq x \leq 5 .$
(ii) Find the exact solutions of $g ^{2021} h (x)=ln 2$, giving your answer in its simplest form.

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