QuestionAugust 24, 2025

What is ((f)/(g))(x) ? f(x)=-x^2+4 g(x)=x^2+9x Write your answer as a polynomial or a rational function in simplest form. square ,xneq square or square

What is ((f)/(g))(x) ? f(x)=-x^2+4 g(x)=x^2+9x Write your answer as a polynomial or a rational function in simplest form. square ,xneq square or square
What is ((f)/(g))(x) ? f(x)=-x^2+4 g(x)=x^2+9x Write your answer as a polynomial or a rational function in simplest form. square ,xneq square or square

Solution
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Answer

\frac{-x^2 + 4}{x^2 + 9x}, x \neq 0, -9 Explanation 1. Define the Rational Function The function (\frac{f}{g})(x) is defined as \frac{f(x)}{g(x)}. 2. Substitute Functions Substitute f(x) = -x^2 + 4 and g(x) = x^2 + 9x into the rational function: \frac{-x^2 + 4}{x^2 + 9x}. 3. Simplify the Expression The expression is already in its simplest form since there are no common factors between the numerator and the denominator. 4. Determine Domain Restrictions Set the denominator x^2 + 9x \neq 0. Solve for x: x(x + 9) \neq 0, so x \neq 0 and x \neq -9.

Explanation

1. Define the Rational Function<br /> The function $(\frac{f}{g})(x)$ is defined as $\frac{f(x)}{g(x)}$.<br /><br />2. Substitute Functions<br /> Substitute $f(x) = -x^2 + 4$ and $g(x) = x^2 + 9x$ into the rational function: $\frac{-x^2 + 4}{x^2 + 9x}$.<br /><br />3. Simplify the Expression<br /> The expression is already in its simplest form since there are no common factors between the numerator and the denominator.<br /><br />4. Determine Domain Restrictions<br /> Set the denominator $x^2 + 9x \neq 0$. Solve for $x$: <br /> $x(x + 9) \neq 0$, so $x \neq 0$ and $x \neq -9$.
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