Find the rules for the composite functions f ∘ g and g ∘ f.
f(x) = 2x2 + 9x + 2; g(x) = x + 3

To find the rules for the composite functions \( f \circ g \) and \( g \circ f \), we will substitute one function into the other.

Given:
- \( f(x) = 2x^2 + 9x + 2 \)
- \( g(x) = x + 3 \)

### 1. Composite Function: \( (f \circ g)(x) \)
The composite function \( f \circ g \) is defined as:
\[
(f \circ g)(x) = f(g(x))
\]

Substitute \( g(x) = x + 3 \) into \( f(x) \):
\[
f(g(x)) = f(x + 3)
\]

Now, compute \( f(x + 3) \) by plugging in \( x + 3 \) for every \( x \) in the function \( f(x) \):

\[
f(x + 3) = 2(x + 3)^2 + 9(x + 3) + 2
\]

Expand and simplify:

\[
(x + 3)^2 = x^2 + 6x + 9
\]

So,

\[
f(x + 3) = 2(x^2 + 6x + 9) + 9(x + 3) + 2
\]

Distribute:

\[
f(x + 3) = 2x^2 + 12x + 18 + 9x + 27 + 2
\]

Combine like terms:

\[
f(x + 3) = 2x^2 + 21x + 47
\]

Thus, the rule for the composite function \( f \circ g \) is:

\[
(f \circ g)(x) = 2x^2 + 21x + 47
\]

### 2. Composite Function: \( (g \circ f)(x) \)
The composite function \( g \circ f \) is defined as:
\[
(g \circ f)(x) = g(f(x))
\]

Substitute \( f(x) = 2x^2 + 9x + 2 \) into \( g(x) \):
\[
g(f(x)) = g(2x^2 + 9x + 2)
\]

Now, compute \( g(2x^2 + 9x + 2) \) by plugging in \( 2x^2 + 9x + 2 \) for every \( x \) in the function \( g(x) \):

\[
g(2x^2 + 9x + 2) = (2x^2 + 9x + 2) + 3
\]

Simplify:

\[
g(2x^2 + 9x + 2) = 2x^2 + 9x + 5
\]

Thus, the rule for the composite function \( g \circ f \) is:

\[
(g \circ f)(x) = 2x^2 + 9x + 5
\]

### Summary of Composite Functions:
- \( (f \circ g)(x) = 2x^2 + 21x + 47 \)
- \( (g \circ f)(x) = 2x^2 + 9x + 5 \)

Would you like further details or have any questions?

---

**Related Questions:**
1. What is the domain of the composite functions \( f \circ g \) and \( g \circ f \)?
2. How do you find the inverse of a composite function?
3. Can you provide a graphical interpretation of these composite functions?
4. How do these compositions change if \( f(x) \) or \( g(x) \) is a different type of function, such as exponential or logarithmic?
5. What are some real-world applications of composite functions?

**Tip:** To verify composite function results, substitute a specific value for \( x \) and check if the calculations for both methods are consistent.

Explore the rules for composite functions f ∘ g and g ∘ f using functions f(x) = 2x^2 + 9x + 2 and g(x) = x + 3. Learn how to compute and interpret these composite functions with detailed step-by-step solutions and examples.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation