https://file.aiquickdraw.com/mathbot/file/17265457823469ia8t5ne.jpg

https://file.aiquickdraw.com/mathbot/file/17265457924406226lfw5.jpg

To solve the problem in the image, let's break it down step by step:

Given:
- \( f(x) = x + 3 \)
- \( g(x) = x - 3 \)

### (a) \( f(g(x)) \)

To find \( f(g(x)) \), we substitute \( g(x) \) into the function \( f(x) \).

1. \( g(x) = x - 3 \), so replace \( x \) in \( f(x) = x + 3 \) with \( g(x) = x - 3 \):
   \[
   f(g(x)) = (x - 3) + 3 = x
   \]
Thus, \( f(g(x)) = x \).

### (b) \( g(f(x)) \)

To find \( g(f(x)) \), we substitute \( f(x) \) into the function \( g(x) \).

1. \( f(x) = x + 3 \), so replace \( x \) in \( g(x) = x - 3 \) with \( f(x) = x + 3 \):
   \[
   g(f(x)) = (x + 3) - 3 = x
   \]
Thus, \( g(f(x)) = x \).

### (c) What is \( g(x) \) called?

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), this means that \( g(x) \) is the **inverse function** of \( f(x) \).

Thus, \( g(x) \) is called an **inverse** function of \( f(x) \).

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Let me know if you want more details on any of the steps!

Here are five related questions to deepen your understanding:

1. What conditions must be met for two functions to be inverses of each other?
2. How can you verify that a function is invertible?
3. What is the graphical relationship between a function and its inverse?
4. Can a function have more than one inverse?
5. How do you find the inverse of more complex functions?

**Tip**: To confirm that two functions are inverses, check if both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true!

If f(x) = x + 3 and g(x) = x - 3, find f(g(x)), g(f(x)), and what g(x) is called.

Learn how to solve f(g(x)) and g(f(x)) given the functions f(x) = x + 3 and g(x) = x - 3. This problem explores inverse functions and key concepts related to function composition. Suitable for high school students in Grades 9-12.

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