Q 24. - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Given information.

We have given,

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

Step 2. Concept used.

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

Step 3. Explanation.

We have,

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

Function g(x) is defined everywhere except x = 0.

So domain of the function is set of all real numbers except x = 0.

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

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

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

Solving it for x,

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

SO domain of the composite function is set of all real numbers except 

x = 0 and $x=-\frac{2}{3}$.

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

Step 4. Conclusion.

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