1. The functions \(f\) and \(g\) are defined by \[\begin{align*} f&: x\mapsto e^x, &&x\in\mathbb{R}, \\ g&: x\mapsto x-1, &&x\in\mathbb{R}. \end{align*}\]

Sketch in a single diagram the graphs of \(f\) and the composite functions \(gf\) and \(fg\), labelling each graph clearly.

How do we compose two functions together?

  1. Express \((1-x)(x-3)\) in the form \(a-(x-b)^2\), where \(a\) and \(b\) are constants. State the coordinates of the maximum point on the graph of \(y=(1-x)(x-3)\), and state what symmetry the graph possesses.

How does expressing \((1-x)(x-3)\) in the form \(a-(x-b)^2\) help us find the maximum value of \(y\)?

Sketch the graph of \(y=e^{(1-x)(x-3)}\).

Things to look out for when sketching a curve:

  • can I find the asymptotes?

  • can I find the intercepts with the axes?

  • can I find the stationary points and their nature?

  • what happens to the curve as \(x\rightarrow \pm \infty ?\)