Question

The function f left parenthesis x right parenthesis equals 2 x plus 5 is​ one-to-one.
a. Find an equation for f Superscript negative 1​, 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 xnot equals
  
enter your response here
C.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for xgreater than or equals
  
enter your response here
D.
f Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here​, for all x

Answer 1

To find the inverse of the function \( f(x) = 2x + 5 \), we'll perform the following steps:

### Part 1: Finding the Inverse Function

1. **Let \( y = f(x) \)**:
\[
y = 2x + 5
\]

2. **Swap \( x \) and \( y \)** to find the inverse:
\[
x = 2y + 5
\]

3. **Solve for \( y \)**:
- Subtract 5 from both sides:
\[
x - 5 = 2y
\]
- Divide by 2:
\[
y = \frac{x - 5}{2}
\]

4. **Write the inverse function**:
\[
f^{-1}(x) = \frac{x - 5}{2}
\]

### Part 2: Verification

Now, let's verify that our inverse is correct by checking the two properties:

1. **Check \( f(f^{-1}(x)) = x \)**:
\[
f(f^{-1}(x)) = f\left(\frac{x - 5}{2}\right) = 2\left(\frac{x - 5}{2}\right) + 5
\]
- Simplifying this:
\[
= (x - 5) + 5 = x
\]

2. **Check \( f^{-1}(f(x)) = x \)**:
\[
f^{-1}(f(x)) = f^{-1}(2x + 5) = \frac{(2x + 5) - 5}{2}
\]
- Simplifying this:
\[
= \frac{2x}{2} = x
\]

Both properties hold, confirming that our inverse function is correct.

### Final Answer

The inverse function is:
\[
f^{-1}(x) = \frac{x - 5}{2}
\]

Now, regarding your options for formatting the inverse function, here is how you can fill out the answer:

**D.**
\[
f^{-1}(x) = \frac{x - 5}{2}, \text{ for all } x
\]