Answer:
Explanation:
Given the functions:
[tex]\begin{gathered} f(x)=-\frac{1}{4x} \\ g(x)=\frac{1}{4x} \end{gathered}[/tex]
In order to know if the functions are inverses of each other, we must show that f(g(x)) is equal to g(f(x)).
Get the composite function f(g(x))
[tex]\begin{gathered} f(g(x))=f(\frac{1}{4x}) \\ \end{gathered}[/tex]
To get f(g(x)), we will replace the variable "x" in f(x) with 1/4x as shown:
[tex]\begin{gathered} f(\frac{1}{4x})=-\frac{1}{4(\frac{1}{4x})} \\ f(\frac{1}{4x})=-\frac{1}{\frac{4}{4x}} \\ f(\frac{1}{4x})=-\frac{1}{\frac{1}{x}} \\ f(\frac{1}{4x})=-x \\ f(g(x))=-x \end{gathered}[/tex]
Next is to get the composite function g(f(x))
[tex]\begin{gathered} g(f(x))=g(-\frac{1}{4x}) \\ \end{gathered}[/tex]
To get g(f(x)), we will replace the variable "x" in g(x) with -1/4x as shown:
[tex]\begin{gathered} g(-\frac{1}{4x})=\frac{1}{4(-\frac{1}{4x})} \\ g(-\frac{1}{4x})=\frac{1}{-\frac{4}{4x}} \\ g(-\frac{1}{4x})=\frac{1}{-\frac{1}{x}} \\ g(-\frac{1}{4x})=-x \\ g(f(x))=-x \end{gathered}[/tex]
From the solution above, since f(g(x)) = g(f(x)) = -x, hence the functions f(x) and g(x) are
