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The image contains a problem asking to determine whether two functions are inverses of each other. The functions provided are \( f(x) = x^2 + 2 \) and \( g(x) = \sqrt{x - 2} \). The problem asks you to:

1. Find \( f(g(x)) \).
2. Find \( g(f(x)) \).
3. Determine if \( g \) is the inverse of \( f \) by evaluating \( f(g(x)) \) and \( g(f(x)) \).

### Let's solve this step by step:

**Step 1: Calculate \( f(g(x)) \):**

Given \( f(x) = x^2 + 2 \) and \( g(x) = \sqrt{x - 2} \), substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(\sqrt{x - 2}) = (\sqrt{x - 2})^2 + 2
\]

Simplify:

\[
f(g(x)) = x - 2 + 2 = x
\]

**Step 2: Calculate \( g(f(x)) \):**

Now, substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(x^2 + 2) = \sqrt{(x^2 + 2) - 2}
\]

Simplify:

\[
g(f(x)) = \sqrt{x^2} = |x|
\]

**Step 3: Determine if \( g \) is the inverse of \( f \):**

- For \( f(g(x)) = x \), this holds true, which is the requirement for \( g \) to be the inverse of \( f \).
- For \( g(f(x)) \), we get \( |x| \). Since \( g(f(x)) \) does not equal \( x \) in all cases (it equals \( x \) if \( x \geq 0 \) and \( -x \) if \( x < 0 \)), \( g(x) \) is **not** the inverse of \( f(x) \).

So, \( g(x) \) is **not** the inverse of \( f(x) \).

Do you want details or have any questions?

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Here are some related questions for further exploration:

1. What are the general steps to verify if two functions are inverses of each other?
2. How does the domain and range of \( f(x) \) and \( g(x) \) affect their relationship as potential inverses?
3. Can a function have more than one inverse? Why or why not?
4. How does the property \( g(f(x)) = |x| \) change the interpretation of \( g(x) \) as an inverse function?
5. What modifications could be made to \( f(x) \) or \( g(x) \) to ensure they are inverses?

**Tip:** Always check both \( f(g(x)) \) and \( g(f(x)) \) to determine if two functions are truly inverses. The relationship must hold for all \( x \) in the domain of the functions.

Determine if the functions f(x) = x^2 + 2 and g(x) = √(x - 2) are inverses by following the steps: (a) Find f(g(x)). (b) Find g(f(x)). (c) Determine if g(x) is the inverse of f(x).

Explore how to verify whether the functions f(x) = x^2 + 2 and g(x) = √(x - 2) are inverses of each other. This detailed solution walks you through the steps of calculating f(g(x)) and g(f(x)), and understanding why g(x) is not the inverse of f(x). Suitable for Grades 10-12.

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