https://file.aiquickdraw.com/mathbot/file/1726257116860niv5rrqn.jpg

Let's analyze the functions given in the image.

We are provided with:

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

### Part (a): \( f(g(x)) \)
To compute \( f(g(x)) \), we substitute \( g(x) = x - 2 \) into the function \( f(x) = x + 2 \):

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

Thus, \( f(g(x)) = x \).

### Part (b): \( g(f(x)) \)
Similarly, for \( g(f(x)) \), substitute \( f(x) = x + 2 \) into the function \( g(x) = x - 2 \):

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

Thus, \( g(f(x)) = x \).

### Part (c): Type of function
Since \( f(g(x)) = x \) and \( g(f(x)) = x \), this means that \( g(x) \) is the **inverse** function of \( f(x) \).

### Summary of answers:
(a) \( f(g(x)) = x \)  
(b) \( g(f(x)) = x \)  
(c) Thus, \( g(x) \) is called an **inverse** function of \( f(x) \).

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Would you like further details or have any questions?

Here are five related questions for practice:
1. How do you verify that two functions are inverses of each other?
2. What is the inverse of \( f(x) = 2x + 3 \)?
3. If \( f(x) = x^2 \), is there an inverse function for all \( x \)?
4. What are the properties of inverse functions?
5. How do you graph a function and its inverse?

**Tip**: When finding the inverse of a function, swap \( x \) and \( y \) in the equation and solve for \( y \).

If f(x) = x + 2 and g(x) = x - 2, find (a) f(g(x)), (b) g(f(x)), and (c) describe g(x) in relation to f(x).

Learn how to compute function compositions and determine that g(x) is the inverse of f(x) when f(x) = x + 2 and g(x) = x - 2. This problem involves key concepts in algebra and inverse functions, suitable for students in Grades 9-12.

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