Problem

The functions $f$ and $g$ are defined by

$$f(x) = \sqrt{x} \quad \text{for } x \geq 0,$$ $$g(x) = 3\sqrt{x + 2} - 5 \quad \text{for } x \geq -2.$$

(a) Describe fully a sequence of transformations which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. You should make clear the order in which the transformations are applied.

The diagram shows the graph of $y = g(x)$.

(b) On the diagram sketch the graph of $y = g^{-1}(x)$ together with any relevant mirror line.

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

(d) State the range of $g^{-1}$.

The function $h$ is defined by

$$h(x) = x - 2 \quad \text{for } x \geq 0.$$

(e) Find the value of $g^{-1}h(4)$.

(f) Explain why the composite function $hg^{-1}$ cannot be formed.