Let's solve the given question step-by-step.
### Part (a): Finding the Inverse Function
We are given the function:
[tex]\[ f(x) = x + 15 \][/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x + 15 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse equation:
[tex]\[ x = y + 15 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x - 15 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = x - 15 \][/tex]
Therefore, the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
### Part (b): Verifying the Inverse Function
We will now verify that the inverse function [tex]\( f^{-1}(x) = x - 15 \)[/tex] is correct by showing that:
1. [tex]\( f(f^{-1}(x)) = x \)[/tex]
2. [tex]\( f^{-1}(f(x)) = x \)[/tex]
1. Verification of [tex]\( f(f^{-1}(x)) = x \)[/tex]:
- Substitute [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f(x - 15) \][/tex]
- Apply the function [tex]\( f \)[/tex]:
[tex]\[ f(x - 15) = (x - 15) + 15 = x \][/tex]
Therefore,
[tex]\[ f(f^{-1}(x)) = x \][/tex]
2. Verification of [tex]\( f^{-1}(f(x)) = x \)[/tex]:
- Substitute [tex]\( f(x) \)[/tex] into [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x + 15) \][/tex]
- Apply the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(x + 15) = (x + 15) - 15 = x \][/tex]
Therefore,
[tex]\[ f^{-1}(f(x)) = x \][/tex]
### Conclusion
Since both verifications hold true, we have confirmed that the inverse function is indeed correct. Thus, the inverse function is:
[tex]\[ f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
And the correct choice is:
[tex]\[ D. \; f^{-1}(x) = x - 15 \; \text{for all } x \][/tex]
