Functions: YIJC 2024 Promo Q7

(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)

$y=0.5$ 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 $f$:

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