Given the definitions of f(x) and g(x) below, find the value of (f∘g)(5)   f(x)=3x^2+6x+15  g(x)=-x+1                                                                                                           

We are given two functions \( f(x) = 3x^2 + 6x + 15 \) and \( g(x) = -x + 1 \), and we need to find \( (f \circ g)(5) \), which means \( f(g(5)) \).

### Step 1: Calculate \( g(5) \)
The function \( g(x) = -x + 1 \), so to find \( g(5) \):

\[
g(5) = -(5) + 1 = -5 + 1 = -4
\]

### Step 2: Calculate \( f(g(5)) = f(-4) \)
Now that we have \( g(5) = -4 \), we substitute \( -4 \) into the function \( f(x) \). The function \( f(x) = 3x^2 + 6x + 15 \), so:

\[
f(-4) = 3(-4)^2 + 6(-4) + 15
\]
\[
= 3(16) + 6(-4) + 15
\]
\[
= 48 - 24 + 15
\]
\[
= 39
\]

### Final Answer:
The value of \( (f \circ g)(5) \) is \( 39 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is \( (g \circ f)(5) \)?
2. How would you find \( (f \circ g)(x) \) in general terms?
3. What is the value of \( f(5) \)?
4. How does the composition of functions work in reverse, i.e., \( (g \circ f)(x) \)?
5. Can you generalize the formula for \( f \circ g \) if both functions are quadratic?

**Tip:** When dealing with compositions of functions, always apply the inner function first, and then substitute the result into the outer function.

Given the definitions of f(x) and g(x) below, find the value of (f∘g)(5). f(x) = 3x^2 + 6x + 15, g(x) = -x + 1.

Learn how to solve a function composition problem involving f(x) = 3x^2 + 6x + 15 and g(x) = -x + 1 to find (f∘g)(5). This example demonstrates step-by-step how to calculate g(5) and then substitute into f(x) for a detailed solution. Suitable for students in Grades 9-11.

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