Question

determine if the given pair of functions are inverse functions of each other using the composition cancellation equations

f(x)=(x - 3)^{2},x\geq3 and g(x)=\sqrt{x}+3

select the correct choice and fill in the answer boxes within your choice

(simplify your answers )

\\( \bigcirc \\) a. f(x) and g(x) are inverse functions of each other because f(g(x))= and g(f(x))=

\\( \bigcirc \\) b. f(x) and g(x) are not inverse functions of each other because f(g(x))= and g(f(x))=

Explanation:

Step1: Calculate \(f(g(x))\)

Substitute \(g(x)=\sqrt{x}+3\) into \(f(x)\).
\(f(g(x)) = (\sqrt{x}+3 - 3)^2\)
Simplify the expression.
\(f(g(x)) = (\sqrt{x})^2=x\)

Step2: Calculate \(g(f(x))\)

Substitute \(f(x)=(x - 3)^2\) into \(g(x)\).
\(g(f(x))=\sqrt{(x - 3)^2}+3\)
Since \(x\geq3\), \(\sqrt{(x - 3)^2}=x - 3\).
\(g(f(x))=(x - 3)+3=x\)

Answer:

A. \(f(x)\) and \(g(x)\) are inverse functions of each other because \(f(g(x)) = x\) and \(g(f(x)) = x\)