Problem

The function $f$ is defined as follows:

$f(x) = \sqrt{x} - 1 \quad \text{for } x > 1.$

(a) Find an expression for $f^{-1}(x)$.

The diagram shows the graph of $y = g(x)$ where $g(x) = \frac{1}{x^2 + 2}$ for $x \in \mathbb{R}$.

(b) State the range of $g$ and explain whether $g^{-1}$ exists.

The function $h$ is defined by $h(x) = \frac{1}{x^2 + 2}$ for $x \geq 0$.

(c) Solve the equation $hf(x) = f\big(\frac{25}{16}\big)$. Give your answer in the form $a + b\sqrt{c}$, where $a$, $b$ and $c$ are integers.