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Functions: YIJC 2024 Promo Q7

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Yishun Innova Junior College2024 PromoPromo Q7●●● Challenging10 marks

What this question tests

Functions, inverses and composition.

Question

The functions $f$ and $g$ are defined by $f:x\mapsto e^x+4$ , $x\in\mathbb R$ , $x\leq1$ , and $g:x\mapsto\sin x$ , $x\in\mathbb R$ , $-\pi<x<\pi$ .

(a) Find $f^{-1}(x)$ and state the domain and range of $f^{-1}$ . [4](b) Determine whether $g^{-1}$ exists. [1]

(c) Explain why the composite function $fg$ exists. [2](d) Find $fg(x)$ and state the domain of $fg$ . [2](e) Determine the exact range of $fg$ . [1]

Show full worked solution▾

(a)Let $y=f(x)$ . $y=e^x+4$ , $e^x=y-4$ , $x=\ln(y-4)$ , $f^{-1}(x)=\ln(x-4)$ .

$D_{f^{-1}}=R_f=(4,4+e]$ , $R_{f^{-1}}=D_f=(-\infty,1]$ .

(b)

$Yishun Innova Junior College 2024 Promo Q7: solution diagram$

$y=0.5$ cuts the graph of $g$ twice (more than once), $g$ is not one-one, $g^{-1}$ does not exist.

(c) $R_g=[-1,1]$ , $D_f=(-\infty,1]$ , $R_g\subseteq D_f$ , hence the composite function $fg$ exists.

(d) $fg(x)=f(\sin x)=e^{\sin x}+4$ , $D_{fg}=D_g=(-\pi,\pi)$ .

(e)To find $R_{fg}$ : Let $R_g=[-1,1]$ be new domain of $f$ :

$Yishun Innova Junior College 2024 Promo Q7: solution diagram$

$R_{fg}=[e^{-1}+4,e+4]$ .

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