The function [tex]$f(x) = x + 6$[/tex] is one-to-one.

a. Find an equation for [tex]$f^{-1}(x)$[/tex], the inverse function.

b. Verify that your equation is correct by showing that [tex]$f\left(f^{-1}(x)\right) = x[tex]$[/tex] and [tex]$[/tex]f^{-1}(f(x)) = x$[/tex].

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. [tex]$f^{-1}(x) = \square[tex]$[/tex], for all [tex]$[/tex]x$[/tex]

B. [tex]$f^{-1}(x) = \square[tex]$[/tex], for [tex]$[/tex]x \neq \square$[/tex]

C. [tex]$f^{-1}(x) = \square[tex]$[/tex], for [tex]$[/tex]x \geq \square$[/tex]

D. [tex]$f^{-1}(x) = \square[tex]$[/tex], for [tex]$[/tex]x \leq \square$[/tex]



Answer :

To solve this problem, we follow these steps:

### a. Finding the Inverse Function

Given the function \( f(x) = x + 6 \), we need to find the inverse function \( f^{-1}(x) \).

1. Set \( y \) equal to \( f(x) \):
[tex]\[ y = x + 6 \][/tex]

2. Solve for \( x \) in terms of \( y \):
[tex]\[ y = x + 6 \implies y - 6 = x \implies x = y - 6 \][/tex]

3. Rewrite the equation with \( x \) and \( y \) switched (since \( y = f(x) \), therefore \( x = f^{-1}(y) \)):
[tex]\[ f^{-1}(x) = x - 6 \][/tex]

### Correct Choice

The correct choice is A.

So, the inverse function is:
[tex]\[ f^{-1}(x) = x - 6 \text{, for all } x \][/tex]

### b. Verifying the Equation

To verify the inverse function is correct, we need to show that:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

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

Calculate \( f(f^{-1}(x)) \):
1. Substitute \( f^{-1}(x) \) into \( f \):
[tex]\[ f(f^{-1}(x)) = f(x - 6) \][/tex]

2. Evaluate \( f(x - 6) \):
[tex]\[ f(x - 6) = (x - 6) + 6 = x \][/tex]

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

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

Calculate \( f^{-1}(f(x)) \):
1. Substitute \( f(x) \) into \( f^{-1} \):
[tex]\[ f^{-1}(f(x)) = f^{-1}(x + 6) \][/tex]

2. Evaluate \( f^{-1}(x + 6) \):
[tex]\[ f^{-1}(x + 6) = (x + 6) - 6 = x \][/tex]

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

### Conclusion

The correct inverse function is:
[tex]\[ f^{-1}(x) = x - 6 \text{, for all } x \][/tex]
This equation satisfies the conditions for an inverse function, as verified by [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex].