Rational Functions
Rational functions are a Grade 7 advanced math topic from Yoshiwara Intermediate Algebra, defined as ratios of two polynomials. Students learn to identify domain restrictions, find vertical and horizontal asymptotes, and analyze the behavior of rational function graphs.
Key Concepts
Property A rational function is a function defined by an algebraic fraction. That is, it has the form $$f(x) = \frac{P(x)}{Q(x)}$$ where $P(x)$ and $Q(x)$ are polynomials. A rational function is undefined at any $x$ values where $Q(x) = 0$. A vertical asymptote is a vertical line on the graph that occurs where a rational function is undefined.
Examples The function $f(x) = \frac{20}{x 5}$ is a rational function. It is undefined when $x=5$, so its graph has a vertical asymptote at the line $x=5$. For the function $g(x) = \frac{x+1}{x^2 9}$, the denominator is zero when $x=3$ or $x= 3$. The graph has two vertical asymptotes: $x=3$ and $x= 3$. The function for average cost $C(n) = \frac{200+8n}{n}$ is rational. It is undefined for $n=0$, which means you cannot produce zero items and calculate a meaningful average cost.
Explanation A rational function is just a fraction made of polynomials. Its graph has a special feature called a vertical asymptote—a vertical line the graph gets very close to but never crosses. This line appears wherever the denominator is zero.
Common Questions
What is a rational function?
A rational function is a function of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
How do you find the domain of a rational function?
The domain includes all real numbers except where the denominator equals zero. Set q(x) = 0 and solve; those values are excluded.
What are vertical asymptotes in a rational function?
Vertical asymptotes occur at x-values where the denominator is zero (and the numerator is not zero). The graph approaches but never reaches these vertical lines.
How do you find the horizontal asymptote of a rational function?
Compare the degrees of numerator and denominator. If degrees are equal, the horizontal asymptote is the ratio of leading coefficients. If the numerator degree is less, the asymptote is y = 0.