Q 21. - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

Step 1. Given information.

We have given functions are,

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

Step 2. Concept used.

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

Step 3. Explanation.

We have,

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

Domain of g(x) is the set of all real numbers except x = 0.

Since, the function g(x) is undefined at x = 0.

Similarly domain of the function f(x) is the set of all real numbers except 

x = 1.

So we need to exclude from the domain from the g(x) that value of x for which g(x) = 1.

$\frac{2}{x}=1$

Solving it for x we will get,

$x=2$

So the domain of the $f\circ g$ is the set of all real numbers except x = 0 and x = 2.

Domain: $(-\infty , 0) \cup (0, 2) \cup (2, \infty )$

Step 4. Conclusion.

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