2.6: Modulus sketch and composite inverse

On the axes below, sketch the graph of $y = |(x + 3)(x - 5)|$ showing the coordinates of the points where the curve meets the $x$-axis.

Write down a suitable domain for the function $f(x) = |(x + 3)(x - 5)|$ such that $f$ has an inverse.

The functions $g$ and $h$ are defined by $g(x) = 3x - 1$ for $x > 1$ and $h(x) = \dfrac{4}{x}$ for $x \neq 0$.

Find $hg(x)$.

Find $(hg)^{-1}(x)$.

Given that $p(a) = b$ and that the function $p$ has an inverse, write down $p^{-1}(b)$.