If  
f
(
x
)
=
x
+
9
  and  
g
(
x
)
=
x
−
9
 ,
(a)  
f
(
g
(
x
)
)
=
  
  
(b)  
g
(
f
(
x
)
)
=
  
  
(c) Thus  
g
(
x
)
  is called an 
 function of  
f
(
x
)
 

Given the functions \( f(x) = x + 9 \) and \( g(x) = x - 9 \), let's solve the parts step by step.

### (a) Find \( f(g(x)) \)
This means we substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(x - 9)
\]

Since \( f(x) = x + 9 \), we replace \( x \) in \( f(x) \) with \( g(x) = x - 9 \):

\[
f(g(x)) = (x - 9) + 9 = x
\]

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

### (b) Find \( g(f(x)) \)
This means we substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g(x + 9)
\]

Since \( g(x) = x - 9 \), we replace \( x \) in \( g(x) \) with \( f(x) = x + 9 \):

\[
g(f(x)) = (x + 9) - 9 = x
\]

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

### (c) Thus, \( g(x) \) is called an **inverse function** of \( f(x) \)
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are inverses of each other.

---

Let me know if you'd like further details or have any questions!

Here are five related questions:
1. What are the properties of inverse functions?
2. How do you prove that two functions are inverses of each other?
3. Can all functions have inverses? If not, why?
4. How do the graphs of inverse functions relate to each other?
5. How does the composition of functions work in general?

**Tip**: If \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( f \) and \( g \) are inverse functions. Always check both compositions!

If f(x) = x + 9 and g(x) = x − 9, (a) f(g(x)) = ? (b) g(f(x)) = ? (c) Thus g(x) is called a function of f(x)

Explore how to find the composition of two functions f(x) = x + 9 and g(x) = x - 9 and learn how g(x) is the inverse function of f(x). This problem introduces the concept of function composition and inverse functions, making it ideal for students in Grades 9-12.

Math Problem Statement

If
f ( x )

x + 9 and
g ( x )

x − 9 , (a)
f ( g ( x ) )

(b)
g ( f ( x ) )

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