Q 22. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$.

Step-by-Step Solution

Answer

Domain of composition of the function $f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$ is $(-\infty , 0) \cup (0, \frac{2}{3}) \cup (\frac{2}{3}, \infty )$.

Step 1. Given information.

We have,

$f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$

Step 2. Concept used.

Domain is the set of all input values where the function is defined and real.

Step 3. Explanation.

We have, 

$f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$

The function g(x) is well defined for all real values of x except x = 0.

So domain of the g(x) is all real numbers except x = 0.

Similarly domain of the f(x) is the set of all real numbers except x = -3.

So we need to exclude from the domain of g(x) that value of x for which $g\left(x\right) = -3$.

So $-\frac{2}{x}=-3$

Solving it for x,

$$\begin{gathered} -2=-3x \\ \Rightarrow x=\frac{2}{3} \end{gathered}$$

That is domain of the composite function of $f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$ is the set of real numbers except x = 0 and $x=\frac{2}{3}$.

Domain: $(-\infty , 0) \cup (0, \frac{2}{3}) \cup (\frac{2}{3}, \infty )$

Step 4. Conclusion.

Hence, domain of composite function of $f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$ is $(-\infty , 0) \cup (0, \frac{2}{3}) \cup (\frac{2}{3}, \infty )$.