4.1 Introduction to Rational Functions
Q: What is a rational function?
A: It is a function of the form:
\[f(x)=\frac{N(x)}{D(x)} \qquad D(x) \neq 0\]
Rational functions can have Vertical and Horizontal Asymptotes
A Vertical Asymptote describes the behavior of a function near a discontinuity. They occur at any \(x\)-value where the numerator IS NOT equal to zero but the denominator IS equal to zero.
Find the domain and vertical asymptotes for:
\[f(x) = \frac{1}{x} \qquad \text{and} \qquad f(x) = \frac{1}{x-3} \qquad \text{and} \qquad f(x) = \frac{1-3x}{x^2+12x+32}\]
A Horizontal Asymptote describes the behavior of a function as \(x\) gets very large. (i.e. What happens to \(y\) as \(x\) goes to \(\infty\)?)
Let \(f\) be the rational function given by:
\[f(x) = \frac{N(x)}{D(x)} = \frac{a_nx^n + a_{n-1}x^{n-1} + \cdots +a_1x + a_0}{b_mx^m + b_{m-1}x^{m-1} + \cdots +b_1x + b_0}\]
where \(N(x)\) and \(D(x)\) have no common factors. The graph of \(f\) has one or no horizontal asymptote determined by comparing the degrees of \(N(x)\) and \(D(x)\).
- If \(n < m\), then the graph of \(f\) has the line \(y = 0\) (the \(x\)-axis) as a horizontal asymptote.
- If \(n=m\) then the graph of \(f\) has the line \(y = \frac{a_n}{b_m}\) as a horizontal asymptote.
- If \(n> m\) then the graph of \(f\) has no horizontal asymptote.
Find the domain of the function and identify any horizontal and vertical asymptotes. Sketch a graph for each.
\(f(x)=\frac{x-4}{(x-2)^2}\)
\(f(x)=\frac{x-4}{1+2x}\)
\(f(x)=\frac{(x-4)^2}{(x+1)^2}\)
\(f(x)=\frac{(x-4)^2}{x-3}\)
\(f(x)=\frac{1}{x} + 2\)
4.2 Graphing Rational Functions
Let:
\[f(x)=\frac{3x^2+4x+1}{3x^2+11x-20}\]
sketch the graph with the:
\(y\)-intercept(s)
\(x\)-intercept(s)
vertical asymptote(s)
horizontal asymptote(s)
Sketch and write an equation for a rational function with:
Vertical asymptotes at \(x = 5\) and \(x = -5\)
\(x\)-intercepts at \(x = 1\) and \(x = 2\)
\(y\) intercept at \(3\)
4.3 Rational Equations and Applications
Rational equalities
Suppose \(f\) varies inversely with \(g\) and that \(f=36\) when \(g=6\). What is the value of \(f\) when \(g=12\)?
Solve the equation:
\[\frac{2}{x} = \frac{4}{3x} -5\]
Solve the equation:
\[\frac{8}{x+1} -\frac{5}{2} = \frac{4}{3x+3}\]
Solve the equation:
\[\frac{x}{2x-4} -9 = \frac{1}{x-2}\]
Solve the equation:
\[\frac{x+1}{x-1} =\frac{-1}{x+3} + \frac{8}{x^2+2x-3}\]
Rational Inequalities
Solve the inequality:
\[\frac{x+4}{x+7} < -3\]
Solve the inequality:
\[\frac{x-2}{x^2-25} < -3\]