f(x)=10x+10f(x)=10x+10

g(x)=x2−9x−10g(x)=x2-9x-10

Find (fg)(x)(fg)(x)

(fg)(x)=(fg)(x)=  

The domain of (fg)(x)(fg)(x) is x≠x≠  

To find \((fg)(x)\), which represents the product of the functions \(f(x)\) and \(g(x)\), we follow these steps:

### Step 1: Multiply the functions

Given:
\[
f(x) = 10x + 10
\]
\[
g(x) = x^2 - 9x - 10
\]

The product \((fg)(x)\) is obtained by multiplying \(f(x)\) and \(g(x)\):
\[
(fg)(x) = f(x) \cdot g(x) = (10x + 10) \cdot (x^2 - 9x - 10)
\]

### Step 2: Expand the expression

Now, expand the product:
\[
(fg)(x) = 10x(x^2 - 9x - 10) + 10(x^2 - 9x - 10)
\]

Expanding each term:
\[
(fg)(x) = 10x^3 - 90x^2 - 100x + 10x^2 - 90x - 100
\]

Combine like terms:
\[
(fg)(x) = 10x^3 - 80x^2 - 190x - 100
\]

### Step 3: Determine the domain of \((fg)(x)\)

The domain of \((fg)(x)\) is determined by considering the domains of \(f(x)\) and \(g(x)\) individually. Both \(f(x)\) and \(g(x)\) are polynomial functions, which means they are defined for all real numbers. Therefore, the domain of \((fg)(x)\) is also all real numbers.

However, if the problem asks for \(x\) values where \((fg)(x)\) is **undefined** or restricted, you might consider potential issues like division by zero. Since there are no such operations here, \((fg)(x)\) has no restrictions, and its domain is:

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

Thus, \((fg)(x)\) is defined for all \(x\), meaning there are no \(x\) values that are excluded from the domain.

### Summary:
- \((fg)(x) = 10x^3 - 80x^2 - 190x - 100\)
- The domain of \((fg)(x)\) is \(x \in \mathbb{R}\) (i.e., all real numbers).

Would you like more details on any part of this solution or have any other questions?

Here are five related questions:
1. How do you find the product of two functions when given their expressions?
2. What changes in the domain if either \(f(x)\) or \(g(x)\) had a denominator?
3. How do you identify restrictions in the domain of a function involving square roots?
4. What would the domain be if \(g(x)\) included a square root of a polynomial?
5. How can we graphically represent the product of two functions?

**Tip:** When multiplying functions, always expand carefully and combine like terms to avoid mistakes in the final expression.

Learn how to calculate the product (f*g)(x) for f(x) = 10x + 10 and g(x) = x^2 - 9x - 10 with a step-by-step solution. This problem involves polynomial functions and their multiplication, determining the domain of the resulting function. Suitable for students in Grades 10-12.

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