9758/2021/P2/Q03

(a) The function $h$ is defined by $h: x \mapsto \frac{1}{2} x^2+3$, for $x \in \mathbf{R}$.
The function $g$ is defined by $g: x \mapsto \frac{x+1}{5 x-1}$, for $x \in \mathbf{R}, x \neq 0.2$.
(i) Find $\operatorname{gh}(2)$.

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(ii) Find the value of $x$ for which $g(x)=1.4$.

(b) The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto \frac{x+a}{2 x+b}$, for $x \in \mathbb{R}, x \neq k$.
(i) Give an expression for $k$ and explain why this value of $x$ has to be excluded from the domain of $f$,

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The function $\mathrm{f}$ is such that $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$.
(ii) Determine the possible values of $a$ and of $b$.

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(iii) Find an expression for $f^{-1}(-4)$.

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