Given that $a$ is a positive constant, describe a sequence of transformations which transform

Question

Given that $a$ is a positive constant, describe a sequence of transformations which transform the graph of $y=e^x$ to the graph of $y=ae^{-x+6}$.[3] The function g and h are defined by $$\begin{align*} \mathrm{g} &: x \mapsto ae^{-x+6}, \quad x \in \mathbb{R}, \quad a > 0, \ \mathrm{h} &: x \mapsto 6 + \ln x, \quad x > 0 \end{align*}$$ Find the range of $g$. [1] Hence determine whether the function hg exists.[1] Given instead that $a=3$. Find $\mathrm{hg}(x)$ and state the domain of $\mathrm{hg}$. [3] Find the exact range of $\mathrm{hg}$. [2]

Accepted Answer

$R_\mathrm{g}=(0,\infty)$; $\mathrm{hg}$ exists; for $a=3$, $\mathrm{hg}(x)=12+\ln3-x$, domain $\mathbb R$

2025 MI PU1 Promo Q7 [Modified]

2025 MI PU1 Promo Q7 [Modified]

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