Answer:
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Step-by-step explanation:
To find the inverse of the function \( g(x) = x + 2 \), denoted as \( g^{-1} \), we follow these steps:
(a) Express \( g^{-1} \) in a similar form.
1. Start with the original function \( g(x) \):
\[ g(x) = x + 2 \]
2. Replace \( g(x) \) with \( y \) (or \( f(x) \)):
\[ y = x + 2 \]
3. Swap \( x \) and \( y \) to solve for \( y \) in terms of \( x \):
\[ x = y + 2 \]
4. Solve this equation for \( y \):
\[ y = x - 2 \]
So, the inverse function \( g^{-1}(x) \) is:
\[ g^{-1}(x) = x - 2 \]
(b) Now, let's evaluate \( g^{-1}(5) \) and \( g^{-1}(-5) \) using the inverse function we found.
(i) For \( g^{-1}(5) \):
\[ g^{-1}(5) = 5 - 2 = 3 \]
(ii) For \( g^{-1}(-5) \):
\[ g^{-1}(-5) = (-5) - 2 = -7 \]
So, the evaluations are:
(i) \( g^{-1}(5) = 3 \)
(ii) \( g^{-1}(-5) = -7 \)
