Find

left parenthesis f circle g right parenthesis left parenthesis x right parenthesis(f◦g)(x)

and

left parenthesis g circle f right parenthesis left parenthesis x right parenthesis(g◦f)(x)

and the domain of each.

​f(x)equals=xplus+11​,

​g(x)equals=3 x squared minus 2 x minus 13x2−2x−1

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Part 1

left parenthesis f circle g right parenthesis left parenthesis x right parenthesis(f◦g)(x)equals=enter your response here

​(Simplify your​ answer.)

To find \((f \circ g)(x)\) and \((g \circ f)(x)\), we will start by defining each function and then proceed with their compositions.

### Step 1: Define the Functions
Given:
- \( f(x) = x + 11 \)
- \( g(x) = 3x^2 - 2x - 1 \)

### Part 1: Find \((f \circ g)(x)\)
The composition \((f \circ g)(x)\) is defined as \(f(g(x))\), meaning we will substitute \(g(x)\) into \(f(x)\).

1. Substitute \(g(x) = 3x^2 - 2x - 1\) into \(f(x)\):
   \[
   f(g(x)) = f(3x^2 - 2x - 1)
   \]
2. Since \(f(x) = x + 11\), replace \(x\) with \(3x^2 - 2x - 1\):
   \[
   f(g(x)) = (3x^2 - 2x - 1) + 11
   \]
3. Simplify the expression:
   \[
   f(g(x)) = 3x^2 - 2x - 1 + 11
   \]
   \[
   f(g(x)) = 3x^2 - 2x + 10
   \]

So, \((f \circ g)(x) = 3x^2 - 2x + 10\).

### Part 2: Find \((g \circ f)(x)\)
The composition \((g \circ f)(x)\) is defined as \(g(f(x))\), meaning we will substitute \(f(x)\) into \(g(x)\).

1. Substitute \(f(x) = x + 11\) into \(g(x)\):
   \[
   g(f(x)) = g(x + 11)
   \]
2. Since \(g(x) = 3x^2 - 2x - 1\), replace \(x\) with \(x + 11\):
   \[
   g(f(x)) = 3(x + 11)^2 - 2(x + 11) - 1
   \]
3. Expand \((x + 11)^2\):
   \[
   (x + 11)^2 = x^2 + 22x + 121
   \]
4. Substitute back:
   \[
   g(f(x)) = 3(x^2 + 22x + 121) - 2(x + 11) - 1
   \]
5. Distribute and simplify each term:
   \[
   = 3x^2 + 66x + 363 - 2x - 22 - 1
   \]
   \[
   = 3x^2 + 64x + 340
   \]

So, \((g \circ f)(x) = 3x^2 + 64x + 340\).

### Part 3: Determine the Domains
- **Domain of \(f(x) = x + 11\):** All real numbers, since it’s a linear function.
- **Domain of \(g(x) = 3x^2 - 2x - 1\):** All real numbers, since it’s a polynomial.

Since both \(f(x)\) and \(g(x)\) are defined for all real numbers, their compositions \((f \circ g)(x)\) and \((g \circ f)(x)\) are also defined for all real numbers.

### Final Answer
1. \((f \circ g)(x) = 3x^2 - 2x + 10\), with domain: all real numbers.
2. \((g \circ f)(x) = 3x^2 + 64x + 340\), with domain: all real numbers.

Would you like further clarification on any of these steps?

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#### Related Questions to Explore
1. What are other properties of function compositions?
2. How does changing \(f(x)\) or \(g(x)\) affect \((f \circ g)(x)\) and \((g \circ f)(x)\)?
3. How would the domains change if \(f(x)\) or \(g(x)\) had restricted domains?
4. Can the composition of functions be commutative? Why or why not?
5. What is the range of \((f \circ g)(x)\) and \((g \circ f)(x)\)?

#### Tip
Always start by carefully identifying each function before substituting; this ensures clarity and prevents errors.

Find (f◦g)(x) and (g◦f)(x) and the domain of each. Given: f(x) = x + 11, g(x) = 3x^2 - 2x - 1.

Explore function composition with step-by-step solutions for finding (f◦g)(x) and (g◦f)(x), and understanding their domains. This problem covers essential algebraic concepts, such as polynomial functions and domain identification. Suitable for students in Grades 10-12.

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