Rational functions
A function that is a ratio of two polynomials is called a rational function. A rational function has the form,
$f(x)=\dfrac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x)\ne0$.
Example: The rational function $f$ is defined as $f(x)=\dfrac{11x+2}{3x^2-48}$. Find $f(2)$ and $f(4)$.
Solution:
Substituting $x=2$ into the function, we get
$f(2)=\dfrac{11\cdot 2+2}{3\cdot 2^2-48}$
$\hphantom{00000}=\dfrac{24}{-36}$
$\hphantom{00000}=\boxed{-\dfrac{2}{3}}$
Substituting $x=4$ into the function, we get
$f(4)=\dfrac{11\cdot 4+2}{3\cdot 4^2-48}$
$\hphantom{00000}=\dfrac{46}{0}$
Since there is a zero in the denominator, $\boxed{f(4)\: is\: undefined.}$
Domain of a rational function
The domain of a rational function is the set of all input values (the $x$ values, if the function is $f(x)$) that do not make the function undefined.
For a rational function $f(x)$, the domain is all real numbers except the $x$ values that make the denominator zero, since a zero denominator makes the function undefined. So, to find the domain, first find the $x$ values (if any) that make the function undefined, and exclude them from the real numbers.
Example 1: Find all the $x$ values for which the expression $\dfrac{3x+5}{8x-16}$ is undefined.
Solution:
To find the $x$ values that make a rational expression undefined, set the denominator equal to zero and solve for $x$.
$8x-16=0$
Solving for $x$, we get $x=2$. So, the answer is $\boxed{x=2}$
Example 2: Find the values of $x$ that are not in the domain of the function,
$g(x)=\dfrac{3x-2}{x^2+3x-10}$
Solution:
Set the denominator of the function $g(x)$ equal to zero and solve for $x$.
$x^2+3x-10=0$
This is a quadratic equation. Solving for $x$, we get
$x=2$ and $x=-5$.
We exclude these $x$ values, since they cannot be in the domain. So the answer is $\boxed{x=2,-5}$.