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?

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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.