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 given or can be inferred 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 identified features to construct the polynomials $p(x)$ and $q(x)$. For example, if a vertical asymptote is at $x=a$, then $(x-a)$ should be a factor of $q(x)$.
4. If x-intercepts are known, include corresponding factors in the numerator $p(x)$. For instance, if there is an x-intercept at $x=b$, then $(x-b)$ should be a factor of $p(x)$.
5. Simplify the function if possible and check that it meets all the given conditions or features of the problem.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the quotient of two polynomials. It takes the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the properties of rational functions, such as their domain, asymptotes, and intercepts, is essential for solving problems related to them.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are foundational in algebra and are used to construct the numerator and denominator of rational functions. Recognizing the degree and leading coefficient of a polynomial helps in analyzing the behavior of the rational function.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator equals zero, while horizontal asymptotes describe the behavior of the function as x approaches infinity. Identifying these asymptotes is crucial for sketching the graph of a rational function and understanding its limits.