If f(x)=4x2+5xf(x)=4x2+5x and g(x)=28−4x2g(x)=28-4x2,

Find and simplify (f+g)(x)(f+g)(x) and state its domain using interval notation

(f+g)(x)=(f+g)(x)=

Domain of (f+g)(x)(f+g)(x) :

Find and simplify (f−g)(x)(f-g)(x) and state its domain using interval notation

(f−g)(x)=(f-g)(x)=

Domain of (f−g)(x)(f-g)(x) :

Find and simplify (fg)(x)(fg)(x) and state its domain using interval notation

(fg)(x)=(fg)(x)=

Domain of (fg)(x)(fg)(x) :

Find (fg)(x)(fg)(x) and state its domain using interval notation

(fg)(x)=(fg)(x)=

Domain of (fg)(x)(fg)(x) :

We are given the functions:
- \( f(x) = 4x^2 + 5x \)
- \( g(x) = 28 - 4x^2 \)

We need to perform several operations on these functions and determine their domains.

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### 1. Find and simplify \( (f + g)(x) \)

\[
(f + g)(x) = f(x) + g(x)
\]

Substitute the given functions:

\[
(f + g)(x) = (4x^2 + 5x) + (28 - 4x^2)
\]

Simplify the expression:

\[
(f + g)(x) = 4x^2 - 4x^2 + 5x + 28 = 5x + 28
\]

#### Domain of \( (f + g)(x) \):
Since \( 5x + 28 \) is a linear function, there are no restrictions on \( x \). The domain is all real numbers:

\[
\text{Domain of } (f + g)(x) = (-\infty, \infty)
\]

---

### 2. Find and simplify \( (f - g)(x) \)

\[
(f - g)(x) = f(x) - g(x)
\]

Substitute the given functions:

\[
(f - g)(x) = (4x^2 + 5x) - (28 - 4x^2)
\]

Simplify the expression:

\[
(f - g)(x) = 4x^2 + 5x - 28 + 4x^2 = 8x^2 + 5x - 28
\]

#### Domain of \( (f - g)(x) \):
Since \( 8x^2 + 5x - 28 \) is a quadratic function, there are no restrictions on \( x \). The domain is all real numbers:

\[
\text{Domain of } (f - g)(x) = (-\infty, \infty)
\]

---

### 3. Find and simplify \( (fg)(x) \)

\[
(fg)(x) = f(x) \cdot g(x)
\]

Substitute the given functions:

\[
(fg)(x) = (4x^2 + 5x)(28 - 4x^2)
\]

Now expand the product:

\[
(fg)(x) = 4x^2(28 - 4x^2) + 5x(28 - 4x^2)
\]

\[
= 4x^2(28) - 4x^2(4x^2) + 5x(28) - 5x(4x^2)
\]

\[
= 112x^2 - 16x^4 + 140x - 20x^3
\]

Rearrange terms in descending powers of \( x \):

\[
(fg)(x) = -16x^4 - 20x^3 + 112x^2 + 140x
\]

#### Domain of \( (fg)(x) \):
Since \( (fg)(x) \) is a polynomial, there are no restrictions on \( x \). The domain is all real numbers:

\[
\text{Domain of } (fg)(x) = (-\infty, \infty)
\]

---

### Summary:

1. \( (f + g)(x) = 5x + 28 \), Domain: \( (-\infty, \infty) \)
2. \( (f - g)(x) = 8x^2 + 5x - 28 \), Domain: \( (-\infty, \infty) \)
3. \( (fg)(x) = -16x^4 - 20x^3 + 112x^2 + 140x \), Domain: \( (-\infty, \infty) \)

---

Would you like further clarification on any part of this process? Here are some related questions for further exploration:

1. What are the roots of \( f(x) + g(x) = 0 \)?
2. How do you factor \( f(x) - g(x) \)?
3. Can you determine the critical points of \( (fg)(x) \)?
4. What is the behavior of \( (fg)(x) \) as \( x \to \infty \)?
5. What is the degree of \( (fg)(x) \)?

**Tip:** Always check the domain for each function before performing any operations, especially when dealing with rational or logarithmic functions!

If f(x) = 4x^2 + 5x and g(x) = 28 - 4x^2, find and simplify (f+g)(x) and state its domain using interval notation. Find and simplify (f−g)(x) and state its domain using interval notation. Find and simplify (fg)(x) and state its domain using interval notation.

Discover how to simplify and find the domains of (f+g)(x), (f−g)(x), and (fg)(x) for given functions f(x) = 4x^2 + 5x and g(x) = 28 - 4x^2. This guide covers polynomial addition, subtraction, and multiplication, and determining the domain in interval notation. Ideal for students in Grades 9-12.

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