Sue Pemberton Solutions for Chapter: Functions, Exercise 5: EXERCISE 2D

The diagram shows the graph of $y=f\left(x\right)$, where $f\left(x\right)=\frac{4}{x+2}$ for $x\in \mathrm{ℝ},x⩾0$.

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

The diagram shows the graph of $y=f\left(x\right)$, where $f\left(x\right)=\frac{4}{x+2}$ for $x\in \mathrm{ℝ},x⩾0$.

(d) On a copy of the diagram, sketch the graph of $y=f^{-1}\left(x\right)$, making clear the relationship between the graphs.

For each of the following functions, find an expression for ${\mathrm{f}}^{-1}\left(x\right)$ and, hence, decide if the graph of $y=\mathrm{f}\left(x\right)$ is symmetrical about the line $y=x$.

(a) $f\left(x\right)=\frac{x+5}{2x-1}$ for $x\in \mathrm{ℝ},x\ne \frac{1}{2}$.

For each of the following functions, find an expression for ${\mathrm{f}}^{-1}\left(x\right)$ and, hence, decide if the graph of $y=\mathrm{f}\left(x\right)$ is symmetrical about the line $y=x$.

(b) $f\left(x\right)=\frac{2x-3}{x-5}$ for $x\in \mathrm{ℝ},x\ne 5$.

For each of the following functions, find an expression for ${\mathrm{f}}^{-1}\left(x\right)$ and, hence, decide if the graph of $y=\mathrm{f}\left(x\right)$ is symmetrical about the line $y=x$.

(c) $f\left(x\right)=\frac{3x-1}{2x-3}$ for $x\in \mathrm{ℝ},x\ne \frac{3}{2}$.

For each of the following functions, find an expression for ${\mathrm{f}}^{-1}\left(x\right)$ and, hence, decide if the graph of $y=\mathrm{f}\left(x\right)$ is symmetrical about the line $y=x$.

(d) $f\left(x\right)=\frac{4x+5}{3x-4}$ for $x\in \mathrm{ℝ},x\ne \frac{4}{3}$.

(a) $f\left(x\right)=\frac{x+a}{bx-1}$ for $x\in \mathrm{ℝ},x\ne \frac{1}{b}$, where $a \&\hspace{0.17em}b$ are constants. Prove that this function is self-inverse.

(b) $g\left(x\right)=\frac{ax+b}{cx+d}$ for $x\in \mathrm{ℝ},x\ne -\frac{d}{c}$, where $a,b,c$ and $d$ are constants. Find the condition for this function to be self-inverse.