Function Families: We've seen three families of functions so far: linear, absolute value, and quadratic.
Today we're going to add another function family to our list: rational functions.
Rational Functions
Definition: a rational function is a quotient of two polynomial functions.
Examples:
$f(x)=\frac{3x^2-2x+5}{x-2}$
$g(x)=\frac{2x+5}{-2x^3-2x^2+5}$
$h(x)=\frac{x^2-2x+5}{1}=x^2-2x+5$ (i.e., all polynomial functions are also rational functions.)
The Domain of Rational Functions
Fact: the domain of any rational function is any real number which doesn't make the denominator 0.
Example: The domain of the rational function $f(x)=\frac{1}{x-2}$ is all real numbers except 2.
We may write the domain as $D=\mathbb{R} \backslash \{2\}$.
Boot vs. Interval Notation
Example: Write the domain of the rational function $g(x)=\frac{1}{x^2-x-6}$ using "boot" notation.
Example: Write the domain of the rational function $h(x)=\frac{2x}{2x-7}$ using "boot" notation.
Example: Write the domain of the rational function $f(x)=\frac{2 x^2+5 x-42}{12 x^2+15 x+3}$ using interval notation.
Graphs of Rational Functions
Example: Consider the graphs of the rational functions below:
$$f(x)=\frac{1}{x-2} \,\,\, \mbox{ and } \,\,\, g(x)=\frac{1}{x^2-x-6}$$.
Graphs of Rational Functions
Vertical Asymptotes: the vertical asymptotes are the vertical lines that the function never crosses.
The function values either approach $\infty$ or $-\infty$.
Fact: the vertical asymptotes of a rational function occur where the denominator is 0 (values of $x$ excluded from the domain).
Rational Expressions
Definition: a rational expression is a quotient of two polynomial expressions.
Examples:
$\frac{3x^2-2x+5}{x-2}$
$\frac{2x+5}{-2x^3-2x^2+5}$
$\frac{x^2-2x+5}{1}=x^2-2x+5$ (i.e., all polynomial expressions are also rational expressions.)
Reducing Rational Expressions
Fact: like regular fractions, we can reduce rational expressions.
Examples: reduce the following rational expressions.
$\frac{x+1}{(x-2)(x+1)}$
$\frac{(2 \theta +5)(3 \theta +1)}{2 \theta +5}$
$\frac{b^2+5b+6}{b^2-b-6}$
Reducing Rational Expressions
Process: To reduce a rational expression:
1) completely factor the numerator and denominator
2) cancel any common factors
Reducing Rational Expressions
More Examples:
$\frac{ 3 \mu^2-19 \mu+20}{\mu^2-5 \mu}$
$\frac{ 49 \alpha^2-49 }{14 + 14 \alpha}$
$\frac{ 4 z^2-37 z+40}{3 z-24}$
$\frac{ 3 u^2-10 u t + 8 t^2}{ 3 u^2-u t -4 t^2}$