Sue Pemberton Solutions for Chapter: Functions, Exercise 11: END-OF-CHAPTER REVIEW EXERCISE 2

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x\in \mathrm{ℝ}$ by

$\mathrm{f}:x\mapsto 2x+1$

$\mathrm{g}:x\mapsto x^2-2$

(iii) Find the value of $b(b\ne a)$ for which $g\left(b\right)=b$.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x\in \mathrm{ℝ}$ by

$\mathrm{f}:x\mapsto 2x+1$

$\mathrm{g}:x\mapsto x^2-2$

(iv) Find and simplify an expression for ${\mathrm{f}}^{-1} \mathrm{g}\left(x\right)$.

The function $h$ is defined by

$\mathrm{h}:x\mapsto x^2-2,\text{ for }x⩽0$.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x\in \mathrm{ℝ}$ by

$\mathrm{f}:x\mapsto 2x+1$

$\mathrm{g}:x\mapsto x^2-2$

(v) Find an expression for $h^{-1}\left(x\right)$.
 

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$f:x\mapsto 2x^2-8x+10$ for $0⩽x⩽2$

$\mathrm{g}:x\mapsto x$ for $0⩽x⩽10$

(i) Express $f\left(x\right)$ in the form $a(x+b{)}^2+c$, where $a,b \& c$ are constants.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$f:x\mapsto 2x^2-8x+10$ for $0⩽x⩽2$

$\mathrm{g}:x\mapsto x$ for $0⩽x⩽10$

(ii) State the range of $\overrightarrow{f}$.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$f:x\mapsto 2x^2-8x+10$ for $0⩽x⩽2$

$\mathrm{g}:x\mapsto x$ for $0⩽x⩽10$

(iii) State the domain of ${\mathrm{f}}^{-1}$.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$f:x\mapsto 2x^2-8x+10$ for $0⩽x⩽2$

$\mathrm{g}:x\mapsto x$ for $0⩽x⩽10$

Sketch on the same diagram the graphs of $y=f\left(x\right), y=g\left(x\right)$ and $y=f^{-1}\left(x\right)$, making clear the relationship between the graphs.

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$f:x\mapsto 2x^2-8x+10$ for $0⩽x⩽2$

$\mathrm{g}:x\mapsto x$ for $0⩽x⩽10$

(v) Find an expression for $f^{-1}\left(x\right)$.