Review question

# Can we sketch the inverse of this composite function? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9824

## Question

The functions $f$, $g$ and $h$ are defined by \begin{align*} f &: x \mapsto \ln x, & \qquad \left(x\in\mathbb{R}^+\right), \\ g &: x \mapsto \frac{1}{x},& \qquad \left(x\in\mathbb{R}^+\right), \\ h &: x \mapsto x^2, &\quad \big(x\in\mathbb{R}\big). \end{align*}
1. Give definitions of each of the functions $fg$ and $f^{-1}$, and state, in each of the following cases, a relationship between the graphs of

1. $f$ and $fg$,

2. $f$ and $f^{-1}$.

The diagram shows a sketch of the graph $y=hf(x)$. State how the sketch shows that $hf$ is not one–one, and prove that, if $\alpha$ and $\beta$, where $0<\alpha <\beta$, are such that $hf(\alpha)=hf(\beta)$, then $\alpha=g(\beta)$.

1. The function $\phi$ is defined by $\phi : x \mapsto hf(x),\qquad (0 < x \leq 1).$ Sketch the graph of $\phi^{-1}$, and give an explicit expression in terms of $x$ for $\phi^{-1}(x)$.