Given that 
f
(
x
)
=
4
x
2
−
5
x
−
2
 and 
g
(
x
)
=
x
+
h
:

Determine the rule for the composite function 
g
(
g
(
x
)
)
.

g
(
g
(
x
)
)
 = 
   

Determine the rule for the composite function 
g
(
f
(
x
)
)
.

g
(
f
(
x
)
)
 = 
   

Determine the rule for the composite function 
f
(
g
(
x
)
)
.

f
(
g
(
x
)
)
 = 

Given the functions:

\[
f(x) = 4x^2 - 5x - 2
\quad \text{and} \quad
g(x) = x + h
\]

Let's determine the rules for the composite functions:

### 1. Composite Function: \( g(g(x)) \)
We know that \( g(x) = x + h \). To find \( g(g(x)) \), substitute \( g(x) \) into \( g \) again:
\[
g(g(x)) = g(x + h) = (x + h) + h = x + 2h
\]
Thus, the rule for \( g(g(x)) \) is:
\[
g(g(x)) = x + 2h
\]

### 2. Composite Function: \( g(f(x)) \)
To find \( g(f(x)) \), substitute \( f(x) = 4x^2 - 5x - 2 \) into \( g \):
\[
g(f(x)) = g(4x^2 - 5x - 2) = (4x^2 - 5x - 2) + h = 4x^2 - 5x - 2 + h
\]
Thus, the rule for \( g(f(x)) \) is:
\[
g(f(x)) = 4x^2 - 5x - 2 + h
\]

### 3. Composite Function: \( f(g(x)) \)
To find \( f(g(x)) \), substitute \( g(x) = x + h \) into \( f \):
\[
f(g(x)) = f(x + h) = 4(x + h)^2 - 5(x + h) - 2
\]
First, expand the terms:
\[
f(g(x)) = 4(x^2 + 2xh + h^2) - 5(x + h) - 2
\]
Simplifying:
\[
f(g(x)) = 4x^2 + 8xh + 4h^2 - 5x - 5h - 2
\]
Thus, the rule for \( f(g(x)) \) is:
\[
f(g(x)) = 4x^2 + 8xh + 4h^2 - 5x - 5h - 2
\]

---

Would you like further details or clarifications?  
Here are 5 related questions to explore:

1. How would the composite functions change if \( h = 0 \)?
2. What is the domain of the composite function \( f(g(x)) \)?
3. How would you graph the function \( f(g(x)) \)?
4. How does the behavior of \( g(g(x)) \) depend on the value of \( h \)?
5. What is the inverse of the function \( g(x) = x + h \)?

**Tip:** Always expand composite functions step by step and simplify carefully to avoid mistakes in handling powers and coefficients.

Given that f(x) = 4x^2 - 5x - 2 and g(x) = x + h:

Determine the rule for the composite function g(g(x)).

Determine the rule for the composite function g(f(x)).

Determine the rule for the composite function f(g(x)).

Explore how to determine composite functions g(g(x)), g(f(x)), and f(g(x)) for quadratic and linear equations. This step-by-step solution includes key algebraic concepts and formulas, making it ideal for high school students in Grades 10-12.

Math Problem Statement

Given that f ( x )

4 x 2 − 5 x − 2 and g ( x )

g ( g ( x ) )

g ( f ( x ) )

f ( g ( x ) )

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