Composition of Function

•If f(x)= 3x + 2x^2 -10 and g(x) = 5x - x +15^2 find:

A. (fg)(x)

C. (g f)(x)

B. (f g)(4)

D. (gf)(6)

To solve the composition of functions problem, we need to clarify each part and carefully apply function composition rules.

### Given:
- \( f(x) = 3x + 2x^2 - 10 \)
- \( g(x) = 5x - x + 15^2 \)

First, simplify \( g(x) \) to:
\[
g(x) = 4x + 225 \quad (\text{since } 15^2 = 225 \text{ and } 5x - x = 4x)
\]

Now, let’s calculate each part:

### A. \((fg)(x)\) or \( f(x) \cdot g(x) \)

This is the product of the two functions:
\[
(fg)(x) = f(x) \cdot g(x)
\]
Substitute the expressions for \( f(x) \) and \( g(x) \):
\[
(fg)(x) = (3x + 2x^2 - 10) \cdot (4x + 225)
\]
Now expand the expression:
\[
(fg)(x) = (3x + 2x^2 - 10)(4x + 225)
= 3x(4x + 225) + 2x^2(4x + 225) - 10(4x + 225)
\]
This simplifies to:
\[
(fg)(x) = 12x^2 + 675x + 8x^3 + 450x^2 - 40x - 2250
\]
Combine like terms:
\[
(fg)(x) = 8x^3 + 462x^2 + 635x - 2250
\]

### B. \((f g)(4)\)

This is the result of applying the function \( f \) to \( g(4) \), i.e., \( f(g(4)) \).

First, find \( g(4) \):
\[
g(4) = 4(4) + 225 = 16 + 225 = 241
\]
Now, apply \( f \) to 241:
\[
f(241) = 3(241) + 2(241)^2 - 10
= 723 + 2(58081) - 10
= 723 + 116162 - 10
= 116875
\]

So, \( (f g)(4) = 116875 \).

### C. \((g f)(x)\)

This is the composition \( g(f(x)) \), meaning applying \( g \) to \( f(x) \). We already know:
\[
f(x) = 3x + 2x^2 - 10
\]
Now apply \( g \) to \( f(x) \):
\[
g(f(x)) = 4(3x + 2x^2 - 10) + 225
= 12x + 8x^2 - 40 + 225
= 8x^2 + 12x + 185
\]

### D. \((g f)(6)\)

This is the result of applying the function \( g \) to \( f(6) \).

First, find \( f(6) \):
\[
f(6) = 3(6) + 2(6)^2 - 10
= 18 + 2(36) - 10
= 18 + 72 - 10 = 80
\]
Now, apply \( g \) to 80:
\[
g(80) = 4(80) + 225
= 320 + 225 = 545
\]

So, \( (g f)(6) = 545 \).

### Final Answers:
- A. \((fg)(x) = 8x^3 + 462x^2 + 635x - 2250\)
- B. \((f g)(4) = 116875\)
- C. \((g f)(x) = 8x^2 + 12x + 185\)
- D. \((g f)(6) = 545\)

Would you like more details on any of these solutions?

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Here are 5 related questions to expand your understanding:

1. What is the general rule for composing two functions \( f \) and \( g \)?
2. How does the composition of functions differ from their product?
3. What is the domain of a composed function \( (f g)(x) \)?
4. How does changing the order of composition affect the result, i.e., \( (f g)(x) \) versus \( (g f)(x) \)?
5. Can function composition always be simplified as easily as in these examples?

**Tip**: When composing functions, always evaluate the inner function first, then use its output as the input for the outer function.

If f(x) = 3x + 2x^2 - 10 and g(x) = 5x - x + 15^2, find:
A. (fg)(x)
B. (f g)(4)
C. (g f)(x)
D. (g f)(6)

Solve the composition of functions problem where f(x) = 3x + 2x^2 - 10 and g(x) = 4x + 225. Learn how to calculate (fg)(x), (f g)(4), (g f)(x), and (g f)(6) step by step, covering polynomial functions and algebraic operations. Suitable for high school students.

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