Chapter 1.9, Problem 32E

a.

To determine

To verify: whether the functions f and g are inverse functions or not algebraically.

Answer to Problem 32E

The functions f and g are inverse functions algebraically.

Explanation of Solution

Given information:

The function $f(x)=\frac{x+3}{x-2},g(x)=\frac{2x+3}{x-1}$

Calculation:

Let $y=f(x)=\frac{x+3}{x-2}$

Find the inverse function $f^{-1}(x)$

  $\begin{array}{l}y=\frac{x+3}{x-2}\\ xy-2y=x+3\\ x(y-1)=3+2y\\ x=\frac{3+2y}{y-1}\\ f^{-1}(y)=\frac{3+2y}{y-1}\\ f^{-1}(x)=\frac{3+2x}{x-1}=g(x)\end{array}$

Hence, the functions f and g are inverse functions algebraically.

a.

To determine

To verify: whether the functions f and g are inverse functions or not graphically.

Answer to Problem 32E

The functions f and g are inverse functions algebraically.

Explanation of Solution

Given information:

The function $f(x)=\frac{x+3}{x-2},g(x)=\frac{2x+3}{x-1}$

Calculation:

Graph the functions $f(x)=\frac{x+3}{x-2},g(x)=\frac{2x+3}{x-1}$

By observing the above graph reflection of one function is another

Function over $y=x$ line.

Hence, the functions f and g are inverse functions algebraically.