Question

The function f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis Superscript 5 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.
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Part 1
a. Select the correct choice below and fill in the answer​ box(es) 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 parenthesisequals
  
enter your response here​, for xless than or equals
  
enter your response here
B.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for all x
C.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xnot equals
  
enter your response here
D.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xgreater than or equals
  
enter your response here

Answer 1

To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = (x + 1)^5 \), we will follow the steps below.

### Part 1: Finding the Inverse Function

1. Start with the equation:
\[
y = (x + 1)^5
\]

2. Swap \( x \) and \( y \) to set up for finding the inverse:
\[
x = (y + 1)^5
\]

3. Solve for \( y \):
- Take the fifth root of both sides:
\[
y + 1 = x^{\frac{1}{5}}
\]
- Then isolate \( y \):
\[
y = x^{\frac{1}{5}} - 1
\]

Thus, the inverse function is:
\[
f^{-1}(x) = x^{\frac{1}{5}} - 1
\]

### Answering the part about the choice:

From the options provided and the function we have found, the correct choice would be:
**B. \( f^{-1}(x) = x^{\frac{1}{5}} - 1 \), for all \( x \)**

### Part 2: Verifying the Inverse

We will now verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

#### Verify \( f(f^{-1}(x)) = x \)

1. Substitute \( f^{-1}(x) \) into \( f(x) \):
\[
f(f^{-1}(x)) = f(x^{\frac{1}{5}} - 1)
\]
2. Compute:
\[
= ((x^{\frac{1}{5}} - 1) + 1)^5
\]
\[
= (x^{\frac{1}{5}})^5
\]
\[
= x
\]

#### Verify \( f^{-1}(f(x)) = x \)

1. Substitute \( f(x) \) into \( f^{-1}(x) \):
\[
f^{-1}(f(x)) = f^{-1}((x + 1)^5)
\]
2. Compute:
\[
= ( (x + 1)^5)^{\frac{1}{5}} - 1
\]
\[
= (x + 1) - 1
\]
\[
= x
\]

Since both conditions are satisfied, we have correctly found the inverse of the function.

### Final Result
The inverse function is as follows:
\[
f^{-1}(x) = x^{\frac{1}{5}} - 1, \text{ for all } x
\]