To find the inverse of the function \( f(x) = x + 2 \), we start by setting \( y = f(x) \):
\[ y = x + 2 \]
Next, we solve for \( x \) in terms of \( y \):
\[ x = y - 2 \]
Now we exchange \( x \) and \( y \) to find the inverse function:
\[ f^{-1}(x) = x - 2 \]
Now, we can compare this result with the given options for \( h(x) \):
- \( h(x) = 18x - 2 \)
- \( h(x) = 9x - 18 \)
- \( h(x) = 9x + 18 \)
- \( h(x) = 18x + 2 \)
None of these functions appear to directly correspond to the inverse function \( f^{-1}(x) = x - 2 \).
Thus, none of the options provided represent the inverse of the function \( f(x) = x + 2 \).