10

10. [Maximum mark: 17]

The function $f$ is defined by $f(x) = 4^x$, where $x \in \mathbb{R}$.

(a) Find $f^{-1}(8)$. Express your answer in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$. [3]

The function $g$ is defined by $g(x) = 1 + \log_2 x$, where $x \in \mathbb{R}^+$.

(b) (i) Find an expression for $g^{-1}(x)$.

(ii) Describe a sequence of transformations that transforms the graph of $y = g^{-1}(x)$ to the graph of $y = f(x)$. [4]

(c) Show that $(f \circ g)(x) = 4x^2$. [3]

The function $h$ is defined by $h(x) = \frac{4x^2}{2x + 1}$, $x \neq -\frac{1}{2}$.

The following diagram shows part of the graph of $h$. Let $R$ be the region enclosed by the graph of $h$ and the $x$‑axis, between the lines $x = 1$ and $x = 3$.
graph of $h(x)$ with shaded region $R$ bounded by $x=1$ and $x=3$

(d) (i) Show that $2x - 1 + \frac{1}{2x + 1} = \frac{4x^2}{2x + 1}$.

(ii) Hence or otherwise, find the area of $R$, giving your answer in the form $p + q \ln r$, where $p, q, r \in \mathbb{Q}^+$. [7]