Let f(x) = x2 + 4x + 10 and g(x) = 3x + 13. Compute the following: (f ∘ f)(x) =

To compute \( (f \circ f)(x) \), it means applying the function \( f(x) \) to itself. In other words, we substitute \( f(x) \) into \( f(x) \).

### Step 1: Recall the given function \( f(x) \)
\[
f(x) = x^2 + 4x + 10
\]

### Step 2: Set up \( (f \circ f)(x) \)
This means we want to find \( f(f(x)) \). We substitute \( f(x) \) into \( f \).

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

### Step 3: Apply the function \( f \) to \( x^2 + 4x + 10 \)
Since \( f(x) = x^2 + 4x + 10 \), we substitute \( x^2 + 4x + 10 \) wherever there is an \( x \) in the formula for \( f(x) \):

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

### Step 4: Simplify the expression
First, we need to expand \( (x^2 + 4x + 10)^2 \):

\[
(x^2 + 4x + 10)^2 = x^4 + 8x^3 + 28x^2 + 80x + 100
\]

Next, expand \( 4(x^2 + 4x + 10) \):

\[
4(x^2 + 4x + 10) = 4x^2 + 16x + 40
\]

Now, combine everything:

\[
f(f(x)) = x^4 + 8x^3 + 28x^2 + 80x + 100 + 4x^2 + 16x + 40 + 10
\]

### Step 5: Combine like terms
\[
f(f(x)) = x^4 + 8x^3 + (28x^2 + 4x^2) + (80x + 16x) + (100 + 40 + 10)
\]
\[
f(f(x)) = x^4 + 8x^3 + 32x^2 + 96x + 150
\]

Thus, the result is:
\[
(f \circ f)(x) = x^4 + 8x^3 + 32x^2 + 96x + 150
\]

Would you like more details on any of the steps? Here are some related questions:

1. What happens when you compose \( g(x) \) with itself?
2. How does function composition differ from function addition?
3. Can you explain how to factor quadratic functions?
4. How would you graph \( f(f(x)) \)?
5. What does the behavior of higher-degree polynomials look like?

**Tip:** When composing functions, ensure each function is substituted into every instance of the variable carefully to avoid missing terms.

Let f(x) = x^2 + 4x + 10 and g(x) = 3x + 13. Compute the following: (f ∘ f)(x) =

Learn how to compute the composition of two functions f(x) = x^2 + 4x + 10 and g(x) = 3x + 13. This detailed solution covers polynomial expansion and function composition, suitable for high school students in Grades 10-12.

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