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].
