Sue Pemberton Solutions for Chapter: Functions, Exercise 3: EXERCISE 2C

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

(c) Find the values of $x$ for which $f\left(x\right)=f^{-1}\left(x\right)$.

Give your answer in surd form.

Determine which of the following functions are self-inverse functions.

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

Determine which of the following functions are self-inverse functions.

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

Determine which of the following functions are self-inverse functions.

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

$f:x\mapsto 3x-5$ for $x\in \mathrm{ℝ} g:x\mapsto 4-2x$ for $x\in \mathrm{ℝ}$

(a) Find an expression for $\left(\mathrm{fg}{)}^{-1}\right(x)$.

$f:x\mapsto 3x-5$ for $x\in \mathrm{ℝ} g:x\mapsto 4-2x$ for $x\in \mathrm{ℝ}$

(b) Find expressions for:

(i) $f^{-1}{g}^{-1}\left(x\right)$

$f:x\mapsto 3x-5$ for $x\in \mathrm{ℝ} g:x\mapsto 4-2x$ for $x\in \mathrm{ℝ}$

(b) Find expressions for:

(i) $f^{-1}{g}^{-1}\left(x\right)$

$f:x\mapsto 3x-5$ for $x\in \mathrm{ℝ} g:x\mapsto 4-2x$ for $x\in \mathrm{ℝ}$

(c) Comment on your results in part b. Investigate if this is true for other functions.