Solution:
(a) Given the functions:
[tex]\begin{gathered} f(x)=x-4 \\ \\ g(x)=x+4 \end{gathered}[/tex]
Then:
[tex]\begin{gathered} f(g(x))=f(x+4) \\ \\ f(x+4)=x+4-4 \\ \\ f(g(x))=x \end{gathered}[/tex]
Similarly,
[tex]\begin{gathered} g(f(x))=g(x-4) \\ \\ g(x-4)=x-4+4 \\ \\ g(f(x))=x \end{gathered}[/tex]
Two functions f and g are inverses of each other if and only if f(g(x))=x for every value of x in the domain of g and g(f(x))=x for every value of x in the domain of f.
ANSWER: f and g are inverse of each other.
(b) Given:
[tex]\begin{gathered} f(x)=-\frac{1}{3x},x0 \\ \\ g(x)=\frac{1}{3x},x0 \end{gathered}[/tex]
Then:
[tex]\begin{gathered} f(g(x))=f(\frac{1}{3x}) \\ \\ f(\frac{1}{3x})=-\frac{1}{3(\frac{1}{3x})} \\ \\ f(g(x))=-x \end{gathered}[/tex]
Also,
[tex]\begin{gathered} g(f(x))=g(-\frac{1}{3x}) \\ \\ g(-\frac{1}{3x})=\frac{1}{3(-\frac{1}{3x})} \\ \\ g(f(x))=-x \end{gathered}[/tex]
ANSWER: f and g are not inverses of each other.
