Chapter 5.1, Problem 21AYU

To determine

To find: The domain of the composite function $f\circ g$ .

Answer to Problem 21AYU

The domain of the composite function $f\circ g$ is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$ .

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=\frac{3}{x-1},\\ g\left(x\right)=\frac{2}{x};\end{array}$

Calculation:

  $\begin{array}{l}f\left(x\right)=\frac{3}{x-1},\\ g\left(x\right)=\frac{2}{x};\end{array}$

Let’s find the domain of the composite $f\circ g$ .

  $\begin{array}{l}f\left(x\right)\circ g\left(x\right)\\ =f\circ g\left(x\right)\\ =f\left(g\left(x\right)\right)\\ =f\left(\frac{2}{x}\right)\\ =\frac{3}{\frac{2}{x}-1}\\ =\frac{3x}{2-x}\end{array}$

Let’s find the domain of the composite $f\circ g$ .

  $\therefore f\circ g=\frac{3x}{2-x}$

For real number output the input should be like that,

Where,

  $\begin{array}{l}2-x\ne 0\\ \Rightarrow x\ne 2\end{array}$

Therefore, domain $=\left(-\infty ,2\right)\cup \left(2,\infty \right)$

Hence, the domain of the composite function $f\circ g$ is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$ .