Question

The function f left parenthesis x right parenthesis equals StartFraction 8 Over x EndFraction
is​ one-to-one.
a. Find an equation for f Superscript negative 1 Baseline left parenthesis x right parenthesis​, the inverse function.
b. Verify that your equation is correct by showing that f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesisequalsx and f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesisequalsx.
Question content area bottom
Part 1
a. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)
A.
f Superscript negative 1 Baseline left parenthesis x right parenthesis equals
  
enter your response here​, for xgreater than or equals
  
enter your response here
B.
f Superscript negative 1 Baseline left parenthesis x right parenthesis equals
  
enter your response here​, for xless than or equals
  
enter your response here
C.
f Superscript negative 1 Baseline left parenthesis x right parenthesis equals
  
enter your response here​, for xnot equals
  
enter your response here
D.
f Superscript negative 1 Baseline left parenthesis x right parenthesis equals
  
enter your response here​, for all x
Part 2
b. Verify that the equation is correct.
f left parenthesis f Superscript negative 1 Baseline left parenthesis x right parenthesis right parenthesis
equals
fleft parenthesis nothing right parenthesis
  
and
f Superscript negative 1 Baseline left parenthesis f left parenthesis x right parenthesis right parenthesis
equals
f Superscript negative 1 Baseline left parenthesis nothing right parenthesis
  
Substitute.
equals
  
enter your response here
equals
  
enter your response here
Simplify.
Part 3
The equation is
▼
not verified.
verified.

Answer 1

To find the inverse of the function \( f(x) = \frac{8}{x} \), we will follow these steps:

### Part 1: Finding the Inverse Function

1. **Start with the equation for the function:**

\[
y = \frac{8}{x}
\]

2. **Swap \( x \) and \( y \) to find the inverse:**

\[
x = \frac{8}{y}
\]

3. **Now, solve for \( y \):**

Multiply both sides by \( y \):

\[
xy = 8
\]

Now, divide both sides by \( x \):

\[
y = \frac{8}{x}
\]

4. **Write the inverse:**

The inverse function is:

\[
f^{-1}(x) = \frac{8}{x}
\]

Since the original function is defined for all \( x \neq 0 \), the same is true for the inverse function.

**Thus, choice D can be selected:**

\[
f^{-1}(x) = \frac{8}{x}, \text{ for } x \neq 0
\]

### Part 2: Verifying the Inverse

Now, we need to verify that:

1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

#### Verification 1: \( f(f^{-1}(x)) \)

Substituting \( f^{-1}(x) \) into \( f(x) \):

\[
f(f^{-1}(x)) = f\left(\frac{8}{x}\right)
\]

Now, substituting into the function \( f \):

\[
f\left(\frac{8}{x}\right) = \frac{8}{\frac{8}{x}} = \frac{8 \cdot x}{8} = x
\]

This means \( f(f^{-1}(x)) = x \).

#### Verification 2: \( f^{-1}(f(x)) \)

Now, substituting \( f(x) \) into \( f^{-1}(x) \):

\[
f^{-1}(f(x)) = f^{-1}\left(\frac{8}{x}\right)
\]

Substituting into the inverse function:

\[
f^{-1}\left(\frac{8}{x}\right) = \frac{8}{\frac{8}{x}} = \frac{8 \cdot x}{8} = x
\]

This means \( f^{-1}(f(x)) = x \).

### Conclusion

Both verifications hold true, confirming that the inverse function is correct.

**The equation is verified.**