Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

Verified step by step guidance

1. Identify the key features of the rational function such as vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts if any are mentioned or can be deduced from the context.
2. Determine the general form of the rational function, which is typically $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials.
3. Use the information about x-intercepts to find the factors of the numerator $p(x)$. Each x-intercept corresponds to a factor of the form $(x-a)$, where $a$ is the x-intercept.
4. Use the information about vertical asymptotes to find the factors of the denominator $q(x)$. Each vertical asymptote corresponds to a factor of the form $(x-b)$, where $b$ is the location of the vertical asymptote.
5. Combine the factors of the numerator and the denominator to form the rational function. Simplify if necessary, and check if the function meets all the given conditions or features.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the structure of rational functions is essential for identifying their properties, such as asymptotes, intercepts, and behavior at infinity.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. They form the numerator and denominator of rational functions, and their degree and coefficients significantly influence the function's graph and behavior.

Finding Equations

Finding an equation for a rational function involves determining the appropriate polynomials for the numerator and denominator based on given conditions or characteristics. This process may include identifying roots, vertical and horizontal asymptotes, and ensuring the function meets specific criteria, such as passing through certain points.