If 
f
(
x
)
=
x
−
9
 and 
g
(
x
)
=
−
5
x
−
3
 , 

Evaluate: 
f
(
g
(
6
)
)
=
 
-39
Incorrect

Find and simplify.  
f
(
g
(
x
)
)
=
 
Correct
  

Find and simplify.  
g
(
f
(
x
)
)
=
 
Correct
  

Find and simplify.  
(
g
∘
g
)
(
x
)
=
 
Incorrect  syntax error: you gave an equation, not an expression.

Evaluate: 
(
f
∘
f
)
(
−
10
)
=
 

Let's go step by step through the problems:

### 1. Find and simplify \( f(g(6)) \):

We are given:
- \( f(x) = x - 9 \)
- \( g(x) = -5x - 3 \)

First, we need to find \( g(6) \):
\[
g(6) = -5(6) - 3 = -30 - 3 = -33
\]
Now, substitute \( g(6) = -33 \) into \( f(x) \):
\[
f(g(6)) = f(-33) = -33 - 9 = -42
\]
So, \( f(g(6)) = -42 \).

### 2. Find and simplify \( f(g(x)) \):

We are looking for \( f(g(x)) \), which means we substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(-5x - 3) = (-5x - 3) - 9 = -5x - 12
\]
Thus, \( f(g(x)) = -5x - 12 \).

### 3. Find and simplify \( g(f(x)) \):

We need to substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(x - 9) = -5(x - 9) - 3 = -5x + 45 - 3 = -5x + 42
\]
Thus, \( g(f(x)) = -5x + 42 \).

### 4. Find and simplify \( (g \circ g)(x) \):

This represents \( g(g(x)) \), which means substituting \( g(x) \) into itself:
\[
g(g(x)) = g(-5x - 3) = -5(-5x - 3) - 3 = 25x + 15 - 3 = 25x + 12
\]
Thus, \( (g \circ g)(x) = 25x + 12 \).

### 5. Evaluate \( (f \circ f)(-10) \):

This represents \( f(f(-10)) \). First, find \( f(-10) \):
\[
f(-10) = -10 - 9 = -19
\]
Now, substitute \( -19 \) back into \( f(x) \):
\[
f(-19) = -19 - 9 = -28
\]
Thus, \( (f \circ f)(-10) = -28 \).

---

### Summary of Results:
1. \( f(g(6)) = -42 \)
2. \( f(g(x)) = -5x - 12 \)
3. \( g(f(x)) = -5x + 42 \)
4. \( (g \circ g)(x) = 25x + 12 \)
5. \( (f \circ f)(-10) = -28 \)

Let me know if you'd like any further details or clarifications. Here are 5 related questions:

1. What is \( f(f(x)) \) in general form?
2. How would you find \( g(g(2)) \)?
3. Can you solve \( f(x) = g(x) \) for \( x \)?
4. What is the inverse of \( f(x) \), if it exists?
5. How do you find \( g^{-1}(x) \) for the given \( g(x) \)?

**Tip:** When dealing with function composition, always substitute the inner function's result into the outer function step by step to avoid mistakes.

If f(x) = x - 9 and g(x) = -5x - 3, evaluate: f(g(6)) = -39 Incorrect. Find and simplify f(g(x)) = Correct. Find and simplify g(f(x)) = Correct. Find and simplify (g ∘ g)(x) = Incorrect syntax error: you gave an equation, not an expression. Evaluate: (f ∘ f)(-10) =

Learn how to evaluate and simplify function compositions with f(x) = x - 9 and g(x) = -5x - 3. This step-by-step guide covers f(g(6)), f(g(x)), g(f(x)), and (f ∘ f)(-10), and explores algebraic concepts related to function composition. Ideal for students in Grades 9-11.

Math Problem Statement

If f ( x )

x − 9 and g ( x )

Evaluate: f ( g ( 6 ) )

Find and simplify.
f ( g ( x ) )

Find and simplify.
g ( f ( x ) )

Find and simplify.
( g ∘ g ) ( x )

Evaluate: ( f ∘ f ) ( − 10 )

Solution

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