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

## Solution

1. Give definitions of each of the functions $fg$ and $f^{-1}$

For $fg$: composing $f$ and $g$ we find $fg(x) = f[g(x)]=\ln\left(\frac{1}{x}\right)=-\ln x.$

For $f^{-1}$: rearranging the formula $y=\ln x$ to make $x$ the subject we find $x=e^y$, so $f^{-1}(x)=e^x.$

…(a) state a relationship between the graphs of $f$ and $fg$

Reflection in the $x$–axis.

…(b) state a relationship between the graphs of $f$ and $f^{-1}.$

Reflection in the line $y=x$.

The diagram shows a sketch of the graph $y=hf(x)$. State how the sketch shows that $hf$ is not one–one.

There are horizontal lines that cut the graph twice. In fact, for every positive $y$-value, there are two $x$-values that $hf$ maps to the $y$-value.

… and prove that, if $\alpha$ and $\beta$, where $0<\alpha <\beta$, are such that $hf(\alpha)=hf(\beta)$, then $\alpha=g(\beta)$.

The composition $hf(x)=h(\ln x)=(\ln x)^2$.

\begin{align*} hf(\alpha) &= hf(\beta) \\ \Longrightarrow(\ln \alpha)^2 &= (\ln\beta)^2\\ \Longrightarrow \ln \alpha &= -\ln\beta. \end{align*} We must have the minus sign here, since $\alpha \neq \beta$. Now we have \begin{align*} \ln \alpha &= -\ln\beta = \ln\left(\frac{1}{\beta}\right)\\ \Longrightarrow \alpha &= \frac{1}{\beta}=g(\beta). \end{align*}

From the graph we see that if $\alpha < \beta$, $0<\alpha < 1$ and $\beta > 1$.

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 $\phi^{-1}(x)$.
\begin{align*} y &= (\ln x)^2 \\ \Longrightarrow -\sqrt{y} &= \ln x \end{align*} (since $0 < x \leq 1$, we know $\ln x$ is negative) and exponentiating both sides gives \begin{align*} x &= e^{-\sqrt{y}} \end{align*}

Switching $x$ and $y$ we find that the inverse of $\phi$ is $\phi^{-1}(x)=e^{-\sqrt{x}}.$ We can find the graph of the inverse of a one-one function by reflecting in the line $y=x$.

Below are the graphs of both $y=\phi(x)$ and $y=\phi^{-1}(x)$.