1.	.Find f◦g and g◦f, where f(x)= x2+1 and g(x)=x+2, are functions from R to R.

To find the compositions \( f \circ g \) and \( g \circ f \) for the functions \( f(x) = x^2 + 1 \) and \( g(x) = x + 2 \), follow these steps:

### Step 1: Find \( f \circ g \) (which is \( f(g(x)) \))

Substitute \( g(x) = x + 2 \) into \( f(x) = x^2 + 1 \):
\[
f(g(x)) = f(x + 2)
\]

Now, substitute \( x + 2 \) into \( f(x) \):
\[
f(x + 2) = (x + 2)^2 + 1
\]

Expand the square:
\[
(x + 2)^2 = x^2 + 4x + 4
\]

So:
\[
f(x + 2) = x^2 + 4x + 4 + 1 = x^2 + 4x + 5
\]

Therefore:
\[
f \circ g(x) = x^2 + 4x + 5
\]

### Step 2: Find \( g \circ f \) (which is \( g(f(x)) \))

Substitute \( f(x) = x^2 + 1 \) into \( g(x) = x + 2 \):
\[
g(f(x)) = g(x^2 + 1)
\]

Now, substitute \( x^2 + 1 \) into \( g(x) \):
\[
g(x^2 + 1) = (x^2 + 1) + 2
\]

Simplify:
\[
g(x^2 + 1) = x^2 + 3
\]

Therefore:
\[
g \circ f(x) = x^2 + 3
\]

### Final Answer:
- The composition \( f \circ g(x) = x^2 + 4x + 5 \)
- The composition \( g \circ f(x) = x^2 + 3 \)

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What happens if you reverse the order of composition in general?
2. How would you find the composition of functions if \( f(x) \) were a different type of function (e.g., exponential)?
3. What is the significance of the domain in the composition of two functions?
4. How do you determine if a composition of functions is one-to-one or onto?
5. Can you find the composition of three functions (e.g., \( h \circ g \circ f \))? What is the process?

**Tip:** When performing function composition, pay close attention to the order since \( f \circ g(x) \) is generally not equal to \( g \circ f(x) \).

Learn how to find the compositions f∘g and g∘f for the functions f(x) = x^2 + 1 and g(x) = x + 2. This problem involves function composition of quadratic and linear functions, covering the process and final expressions step-by-step. Suitable for Grades 10-12 students studying function compositions.

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