A. Find (fg)(x) for each pair of functions. (12 points)

1.f(x)=2x+6 and g(x)=3x-3

Solution and answer:

2. f(x)= 3x and g(x)=x²-2x+1

Solution and answer:

3. f(x)=3x+6 and g(x)=2x²+x+4

Solution and answer:

4. f(x)=x²-3x+2 and g(x)=x²-2x+1

Solution and answer:

To find \((fg)(x)\) for each pair of functions, you need to multiply the two functions together. That is, \((fg)(x) = f(x) \cdot g(x)\). Below are the solutions and answers for each pair:

### 1. \(f(x) = 2x + 6\) and \(g(x) = 3x - 3\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (2x + 6) \cdot (3x - 3)
\]

Expand the product:

\[
(fg)(x) = 2x \cdot 3x + 2x \cdot (-3) + 6 \cdot 3x + 6 \cdot (-3)
\]

\[
(fg)(x) = 6x^2 - 6x + 18x - 18
\]

Combine like terms:

\[
(fg)(x) = 6x^2 + 12x - 18
\]

**Answer:** \((fg)(x) = 6x^2 + 12x - 18\)

---

### 2. \(f(x) = 3x\) and \(g(x) = x^2 - 2x + 1\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = 3x \cdot (x^2 - 2x + 1)
\]

Distribute \(3x\) across each term in \(g(x)\):

\[
(fg)(x) = 3x \cdot x^2 + 3x \cdot (-2x) + 3x \cdot 1
\]

\[
(fg)(x) = 3x^3 - 6x^2 + 3x
\]

**Answer:** \((fg)(x) = 3x^3 - 6x^2 + 3x\)

---

### 3. \(f(x) = 3x + 6\) and \(g(x) = 2x^2 + x + 4\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (3x + 6) \cdot (2x^2 + x + 4)
\]

Expand the product:

\[
(fg)(x) = 3x \cdot 2x^2 + 3x \cdot x + 3x \cdot 4 + 6 \cdot 2x^2 + 6 \cdot x + 6 \cdot 4
\]

\[
(fg)(x) = 6x^3 + 3x^2 + 12x + 12x^2 + 6x + 24
\]

Combine like terms:

\[
(fg)(x) = 6x^3 + 15x^2 + 18x + 24
\]

**Answer:** \((fg)(x) = 6x^3 + 15x^2 + 18x + 24\)

---

### 4. \(f(x) = x^2 - 3x + 2\) and \(g(x) = x^2 - 2x + 1\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (x^2 - 3x + 2) \cdot (x^2 - 2x + 1)
\]

Expand the product:

\[
(fg)(x) = x^2 \cdot (x^2 - 2x + 1) - 3x \cdot (x^2 - 2x + 1) + 2 \cdot (x^2 - 2x + 1)
\]

Now expand each term:

\[
(fg)(x) = (x^4 - 2x^3 + x^2) - (3x^3 - 6x^2 + 3x) + (2x^2 - 4x + 2)
\]

Combine like terms:

\[
(fg)(x) = x^4 - 5x^3 + 9x^2 - 7x + 2
\]

**Answer:** \((fg)(x) = x^4 - 5x^3 + 9x^2 - 7x + 2\)

---

### Questions to Expand Understanding:

1. How does the degree of the resulting polynomial relate to the degrees of the original functions?
2. Can the product \((fg)(x)\) ever result in a polynomial of lower degree than the original functions?
3. What is the impact of a zero coefficient in one of the terms of the functions \(f(x)\) or \(g(x)\) on \((fg)(x)\)?
4. How would the result change if \(f(x)\) or \(g(x)\) were a constant function?
5. What would happen if you composed the functions instead of multiplying them? How would \((f \circ g)(x)\) differ from \((fg)(x)\)?

### Tip:

When multiplying polynomials, always take care to distribute each term properly and then combine like terms to simplify the expression.

Learn how to compute (fg)(x) for each pair of functions: f(x)=2x+6 and g(x)=3x-3. Detailed solutions provided for polynomial multiplication and function composition. Suitable for advanced high school students and those studying function transformations.

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