Question

For the three-part question that follows, provide your answer to each question in the given workspace. Identify each part with a coordinating response. Be sure to clearly label each part of your response as PartA, Part B, and Part C.Use the information below for Part A Part B, and Part C.

Answer 1

PART A:

In order to calculate (fog)(x), that is, f(g(x)), we need to use g(x) as the input value (value of x) of f(x).

So we have:

$\begin{gathered} f(x)=\frac{1}{2}x-7 \\ f(g(x))=\frac{1}{2}g(x)-7 \\ f(g(x))=\frac{1}{2}(2x+14)-7 \\ f(g(x))=x+7-7 \\ f(g(x))=x \end{gathered}$

PART B:

Let's use f(x) as the input value of g(x):

$\begin{gathered} g(x)=2x+14 \\ g(f(x))=2f(x)+14 \\ g(f(x))=2(\frac{1}{2}x-7)+14 \\ g(f(x))=x-14+14 \\ g(f(x))=x \end{gathered}$

PART C:

When two functions are inverse to each other, we have the following property:

$f(f^{-1}(x))=f^{-1}\mleft(f\mleft(x\mright)\mright)=x$

Since the composite functions of f(x) and g(x) are equal to x, therefore f(x) and g(x) are inverse functions.