Given: Two functions as follows
[tex]\begin{gathered} f(x)=16x^2 \\ g(x)=\frac{1}{4}\sqrt{x} \end{gathered}[/tex]
Required: To find f(g(x)) and g(f(x)).
Explanation: f(g(x)) can be determined by putting g(x) in f(x) as follows
[tex]f(g(x))=16[g(x)]^2[/tex][tex]\begin{gathered} f(g(x))=16(\frac{1}{4}\sqrt{x})^2 \\ =16\times\frac{x}{16} \\ =x \end{gathered}[/tex]
Similarly, for g(f(x)) we have
[tex]g(f(x))=\frac{1}{4}(\sqrt{16x^2})[/tex][tex]\begin{gathered} =\frac{1}{4}\times4x \\ =x \end{gathered}[/tex]
Since. f(g(x))=g(f(x)) the given functions f(x) and g(x) are inverse.
Final Answer: a) f(g(x)) is x
b) g(f(x)) is x
c) Functions f and g are inverse functions.
