The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{2}{{e}^{1-{

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{2}{{e}^{1-{{x}^{2}}}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x\le 1$ $\mathrm{g}:x\mapsto \sqrt{1-\ln x},\mathrm{ }x\in \mathbb{R},\mathrm{ }0 Show that $\mathrm{gf}$ exists, and find the range of $\mathrm{gf}$. [4] Justify, with a reason, whether ${{\mathrm{f}}^{-1}}$ exists. [2] The domain of $\mathrm{f}$ is restricted to $\left( -\infty ,b \right]$ such that $b$ is the largest value for which the inverse function ${{\mathrm{f}}^{-1}}$ exists. State the value of $b$ and define ${{\mathrm{f}}^{-1}}$clearly. [4] The graph of $y=\mathrm{h}\left( x \right)$ is obtained by transforming the graph of $y=\mathrm{g}(x)$ in the following $2$ steps. Step 1: Scale parallel to the $x$-axis by a factor of $2$. Step 2: Reflect in the $x$-axis. Define $\mathrm{h}$ in a similar form. [3]

Accepted Answer

gf exists, R_gf = [√(ln2), ∞); f⁻¹ doesn't exist; b = 0, f⁻¹(x) = −√(1−ln(2x)); h(x) = −√(1−ln(x/2))

2016 HCI P1 Q11

2016 HCI P1 Q11

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