f(x)=x²+5x+6.     g(x)=x+2 solve this problem using composition function

To solve the problem using composition functions, we need to find \( (f \circ g)(x) \) and \( (g \circ f)(x) \). 

### 1. \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means \( f(g(x)) \). This means we substitute \( g(x) \) into the function \( f(x) \).

Given:
\[
f(x) = x^2 + 5x + 6
\]
\[
g(x) = x + 2
\]

So, \( f(g(x)) = f(x + 2) \).

Now, substitute \( x + 2 \) into \( f(x) \):

\[
f(x + 2) = (x + 2)^2 + 5(x + 2) + 6
\]

Let's expand this:

\[
(x + 2)^2 = x^2 + 4x + 4
\]

\[
5(x + 2) = 5x + 10
\]

So:

\[
f(x + 2) = x^2 + 4x + 4 + 5x + 10 + 6
\]

Combine like terms:

\[
f(x + 2) = x^2 + 9x + 20
\]

Therefore:

\[
(f \circ g)(x) = x^2 + 9x + 20
\]

### 2. \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means \( g(f(x)) \). This means we substitute \( f(x) \) into the function \( g(x) \).

Given:
\[
g(x) = x + 2
\]

Now substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = f(x) + 2
\]

Substitute \( f(x) = x^2 + 5x + 6 \):

\[
g(f(x)) = (x^2 + 5x + 6) + 2
\]

Simplify:

\[
g(f(x)) = x^2 + 5x + 8
\]

Therefore:

\[
(g \circ f)(x) = x^2 + 5x + 8
\]

### Summary
- \( (f \circ g)(x) = x^2 + 9x + 20 \)
- \( (g \circ f)(x) = x^2 + 5x + 8 \)

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would you solve \( (f \circ g)(2) \)?
2. Can you find the inverse of \( f(x) \) and \( g(x) \)?
3. What happens when you compose a function with its inverse?
4. How does the order of composition affect the outcome?
5. How do you find the domain of the composed function \( (f \circ g)(x) \)?

**Tip:** When composing functions, always substitute one function completely into the other before simplifying.

Explore the composition of functions f(x)=x²+5x+6 and g(x)=x+2. Learn how to solve problems involving function composition, including (f ∘ g)(x) and (g ∘ f)(x). Suitable for students in Grades 10-12.

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