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.