Find f left parenthesis g left parenthesis x right parenthesis right parenthesis and g left parenthesis f left parenthesis x right parenthesis right parenthesis and determine whether the pair of functions f and g are inverses of each other. Complete parts a through c below.

Question

f left parenthesis x right parenthesis equals 17 x and g left parenthesis x right parenthesis equals StartFraction x Over 17 EndFraction
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Part 1
a.  f left parenthesis g left parenthesis x right parenthesis right parenthesisequals
  
enter your response here  ​(Simplify your​ answer.)
Part 2
b.  g left parenthesis f left parenthesis x right parenthesis right parenthesisequals
  
enter your response here ​ (Simplify your​ answer.)
Part 3
c. Are the two functions f and g inverses of each​ other?
f and g are inverses of each other.
f and g are not inverses of each other.

Answer 1

Let's start by calculating $f(g(x))$ and $g(f(x))$ for the given functions $f(x)=17x$ and $g(x)=\frac{x}{17}$.

Part 1: Find $f(g(x))$

To find $f(g(x))$:

1. Substitute $g(x)$ into $f(x)$: $$g(x)=\frac{x}{17}$$ Then, $$f(g(x))=f\left(\frac{x}{17}\right)=17\left(\frac{x}{17}\right)$$
2. Simplify: $$f\left(\frac{x}{17}\right)=17\cdot \frac{x}{17}=x$$

Thus,
$$\boxed{{\displaystyle x}}$$

Part 2: Find $g(f(x))$

To find $g(f(x))$:

1. Substitute $f(x)$ into $g(x)$: $$f(x)=17x$$ Then, $$g(f(x))=g(17x)=\frac{17x}{17}$$
2. Simplify: $$g(17x)=17x\cdot \frac{1}{17}=x$$

Thus,
$$\boxed{{\displaystyle x}}$$

Part 3: Determine if $f$ and $g$ are inverses

Since we have found:

• $f(g(x))=x$
• $g(f(x))=x$

This shows that: $$f(g(x))=x\text{and}g(f(x))=x$$

Therefore, $f$ and $g$ are inverses of each other.

Thus, the answer is:
$$\text{f and g are inverses of each other.}$$

You can conclude with:
$$\text{f and g are inverses of each other.}$$