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

Question

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.

State briefly the relationship

1. between the graphs of $f$ and $gf$,

2. between the graphs of $f$ and $fg$.

2. 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.

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