f(x) = x2 − x and g(x) = 17 − x2.    (fg)1/2



To solve the expression \((fg)^{1/2}\), where \( f(x) = x^2 - x \) and \( g(x) = 17 - x^2 \), we need to follow these steps:

### Step 1: Find the product of \( f(x) \) and \( g(x) \).

\[
f(x) \cdot g(x) = (x^2 - x) \cdot (17 - x^2)
\]

### Step 2: Expand the product.

\[
f(x) \cdot g(x) = x^2(17 - x^2) - x(17 - x^2)
\]
\[
f(x) \cdot g(x) = 17x^2 - x^4 - 17x + x^3
\]
\[
f(x) \cdot g(x) = -x^4 + x^3 + 17x^2 - 17x
\]

### Step 3: Take the square root of the product.

\[
(fg)^{1/2} = \sqrt{-x^4 + x^3 + 17x^2 - 17x}
\]

This is the expression for \((fg)^{1/2}\).

Would you like more details or have any questions about the process? Here are some related questions for further practice:

1. What is the domain of the function \((fg)^{1/2}\)?
2. How would you find the zeros of the function \( f(x) \)?
3. What is the maximum value of \( g(x) \)?
4. How do you find the derivative of the product \( f(x) \cdot g(x) \)?
5. How would you determine if \( f(x) \cdot g(x) \) has any points of inflection?

**Tip:** When dealing with square roots of polynomials, always check for where the expression under the square root is non-negative, as this determines the domain of the square root function.

f(x) = x^2 − x and g(x) = 17 − x^2. (fg)^{1/2}

Learn how to solve the expression (fg)^{1/2}, where f(x) = x^2 − x and g(x) = 17 − x^2, by expanding the product and taking the square root. This problem covers key algebraic concepts and is suitable for students in Grades 9-11.

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