Review question

# Can we sketch and describe these composite functions? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5595

## Suggestion

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 $x\rightarrow \pm \infty ?$