Match each type of function with a description of its domain. Prompts $$ \begin{array}{ll} ext { (1) polynomial function } & ext { Select match } \ ext { (2) rational function } & \begin{array}{l} ext { The domain is all real numbers. } \ ext { denominator zero. }\end{array} \ \begin{array}{ll} ext { (3) square root function } & \begin{array}{l} ext { The domain is all real numbers except those that make the radicand } \ ext { negative. }\end{array}\end{array}\end{array} $$

Question

Match each type of function with a description of its domain. Prompts $$ \begin{array}{ll}	ext { (1) polynomial function } & 	ext { Select match } \ 	ext { (2) rational function } & \begin{array}{l}	ext { The domain is all real numbers. } \ 	ext { denominator zero. }\end{array} \ \begin{array}{ll}	ext { (3) square root function } & \begin{array}{l}	ext { The domain is all real numbers except those that make the radicand } \ 	ext { negative. }\end{array}\end{array}\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Here's how each type of function matches with its description of the domain:

1. **Polynomial Function**
   - **Domain:** All real numbers.
   
   **Explanation:** Polynomial functions, such as $$ f(x) = x^2 + 3x + 2 $$, are defined for every real number. There are no restrictions on the values that $$ x $$ can take.

2. **Rational Function**
   - **Domain:** All real numbers except those that make the denominator zero.
   
   **Explanation:** Rational functions are of the form $$ f(x) = \frac{P(x)}{Q(x)} $$, where $$ P(x) $$ and $$ Q(x) $$ are polynomials. The function is undefined when the denominator $$ Q(x) = 0 $$, so those values must be excluded from the domain.

3. **Square Root Function**
   - **Domain:** All real numbers except those that make the radicand negative.
   
   **Explanation:** Square root functions are of the form $$ f(x) = \sqrt{g(x)} $$. The expression inside the square root, called the radicand $$ g(x) $$, must be non-negative because the square root of a negative number is not a real number. Therefore, $$ g(x) \geq 0 $$ defines the domain.

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**Summary of Matches:**

1. **Polynomial Function** – The domain is all real numbers.
2. **Rational Function** – The domain is all real numbers except those that make the denominator zero.
3. **Square Root Function** – The domain is all real numbers except those that make the radicand negative.
- **Polynomial Function:** All real numbers. - **Rational Function:** All real numbers except where the denominator is zero. - **Square Root Function:** All real numbers except where the radicand is negative.

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