The functions \(f\), \(g\) and \(h\) are defined by \[\begin{align*} f &: x \mapsto \ln x, & \qquad \left(x\in\mathbb{R}^+\right), \\ g &: x \mapsto \frac{1}{x},& \qquad \left(x\in\mathbb{R}^+\right), \\ h &: x \mapsto x^2, &\quad \big(x\in\mathbb{R}\big). \end{align*}\]
  1. Give definitions of each of the functions \(fg\) and \(f^{-1}\), and state, in each of the following cases, a relationship between the graphs of

    1. \(f\) and \(fg\),

    2. \(f\) and \(f^{-1}\).

Graph of y = hf(x) against x for positive x. The graph approaches infinity as x tends to zero and is increasing for larger x, with a minimum at (1, 0).

The diagram shows a sketch of the graph \(y=hf(x)\). State how the sketch shows that \(hf\) is not one–one, and prove that, if \(\alpha\) and \(\beta\), where \(0<\alpha <\beta\), are such that \(hf(\alpha)=hf(\beta)\), then \(\alpha=g(\beta)\).

  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 \(\phi^{-1}(x)\).