Review question

# Can we work with the domain, codomain and range of a bijection? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5966

## Question

1. The function $f$ from $\mathbb{R}$ to $\mathbb{R}$ is given by $f: x\mapsto x^2+2x+2$. Find the range of $f$, and state, with a reason, whether or not $f$ is bijective.

The range of $f$ is denoted by $S$, and the function $g$ from $S$ to $\mathbb{R}$ is given by $g: x\mapsto 1/x$. State the image of $x$ under the composite function $g\circ f$, and give the range of this composite function.

2. $R'$ denotes the set of real numbers excluding $0$ and $1$. Functions $\phi$ and $\psi$ from $R’$ to $R'$ are given by $\phi : x\mapsto \frac{1}{1-x}, \qquad \qquad \psi : x \mapsto 1-x.$

Give in a similar form the definitions of $\phi ^{-1}$ and $(\phi \circ \psi)^{-1}$.