Q 23. - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$

Step-by-Step Solution

Answer

The domain of the composite function of $f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$ is

$(-\infty , -4) \cup (-4, 0) \cup (0, \infty )$.

Step 1. Given information.

We have given,

$f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$

Step 2. Concept used.

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

Step 3. Explanation.

We have,

$f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$

The function g(x) is defined for all real numbers except x = 0.

So domain of the g(x) is the set of all real values of x except 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 of g(x) that value of x for which $g\left(x\right) = 1$.

That is,

$-\frac{4}{x}=1$

Solving it for x,

So domain of the composite function is the set of all real numbers except $x = -4 and x = 0$.

Domain: $(-\infty , -4) \cup (-4, 0) \cup (0, \infty )$.

Step 4. Conclusion.

Hence domain of the composite function of $f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$ is $(-\infty , -4) \cup (-4, 0) \cup (0, \infty )$.