Chapter 1.5, Problem 32E

Solution

To verify: The functions f and g are inverses by showing $f\left(g\left(x\right)\right)=x$ , and $g\left(f\left(x\right)\right)=x$ .

Given data: The functions $f\left(x\right)$ and $g\left(x\right)$ are; $f\left(x\right)=\frac{x+3}{x-2}$ and $g\left(x\right)=\frac{2x+3}{x-1}$ .

Method/Formula used:

The composite functions of two functions $f\left(x\right)$ and $g\left(x\right)$ are denoted by $\left(fog\right)\left(x\right)$ and $\left(gof\right)\left(x\right)$ given by $\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)$ , and $\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)$ .

Verification:

The given functions are; $f\left(x\right)=\frac{x+3}{x-2}$ and $g\left(x\right)=\frac{2x+3}{x-1}$ . Therefore,

  $\begin{array}{c}f\left(g\left(x\right)\right)=f\left(g\left(x\right)\right)\leftarrow \text{substitute }\left(\frac{2x+3}{x-1}\right)\text{for}g\left(x\right)\\ =f\left(\frac{2x+3}{x-1}\right)\text{}\text{}\text{}\text{}\leftarrow \text{use}\text{definition}\text{of}f\left(x\right)=\frac{x+3}{x-2}\\ =\frac{\left(\frac{2x+3}{x-1}\right)+3}{\left(\frac{2x+3}{x-1}\right)-2}\leftarrow \text{simplify}\\ =\frac{\frac{\left(2x+3\right)+\left(3x-3\right)}{x-1}}{\frac{\left(2x+3\right)-2x+2}{x-1}}\\ =\frac{5x+\cancel{3}-\cancel{3}}{3+2}\\ =x\end{array}$

Further,

  $\begin{array}{c}g\left(f\left(x\right)\right)=g\left(f\left(x\right)\right)\leftarrow \text{substitute}\left(\frac{x+3}{x-2}\right)\text{for}f\left(x\right)\\ =g\left(\frac{x+3}{x-2}\right)\leftarrow \text{use}\text{definition}\text{of}g\left(x\right)=\frac{2x+3}{x-1}\\ =\frac{2\left(\frac{x+3}{x-2}\right)+3}{\left(\frac{x+3}{x-2}\right)-1}\leftarrow \text{simplify}\\ =\frac{2\left(x+3\right)+3\left(x-2\right)}{\left(x+3\right)-\left(x-2\right)}\\ =\frac{2x+\cancel{6}+3x-\cancel{6}}{\cancel{x}-\cancel{x}+3+2}\\ =\frac{5x}{5}\\ =x\end{array}$

Now, $f\left(g\left(x\right)\right)=x\Rightarrow g=f^{-1}\left(x\right)$ , i.e., the function g is the inverse of function f .

Also, $g\left(f\left(x\right)\right)=x\Rightarrow f=g^{-1}\left(x\right)$ , i.e., the function f is the inverse of function g .

That shows the function f and g are inverses.