1. Give definitions of each of the functions \(fg\) and \(f^{-1}\) and state the relationship between the graphs of \(f\) and both \(fg\) and \(f^{-1}\).

The function \(\ln x\) is the logarithm to base \(e\) of \(x\).

Be careful with the order of your functions when composing them! If we take the function \(fg\), which order for \(f\) and \(g\) do we apply here?

Would a sketch help us describe the relationship between \(f\) and \(fg\)? And \(f\) and \(f^{-1}\)?

  1. State how the sketch shows that \(hf\) is not one–one…

What does one–one mean? What do graphs of one–one functions look like?

…and prove that, if \(\alpha\) and \(\beta\), where \(0<\alpha <\beta\), are such that \(hf(\alpha)=hf(\beta)\), then \(\alpha=g(\beta)\).

What is the function \(hf\)?

Sketch the graph of \(\phi^{-1}\), and give an explicit expression in terms of \(x\) for \(\phi^{-1}(x)\).

How do we find the inverse of a function? And its graph?

What happens if we sketch \(y=\phi(x)\) on a graph, then the line \(y=x\)?

How do we find the inverse of a function using algebra?

Can we rearrange \(y=(\ln x)^2\) to make \(x\) the subject? What happens when we take the square root?