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 $\boxed{x=2,-5}$.
Once we have found the excluded values, we can write the domain of the function. There are two common ways to write it.
Interval notation describes the domain using intervals of the number line. Removing the excluded values splits the number line into intervals, which we join with the union symbol $\cup$. A parenthesis, $($ or $)$, means the endpoint is not included.
Set-builder notation describes the domain with a rule. For example, $\{x\mid x\text{ is a real number},\ x\ne 3\}$ is read as "the set of all $x$ such that $x$ is a real number and $x$ is not equal to $3$." The bar $\mid$ means "such that."
Example 3: Find the domain of the rational function $f(x)=\dfrac{7x-8}{13x+25}$.
Solution:
Find the excluded values by setting the denominator equal to zero and solving for $x$:
$13x+25=0$
Subtract $25$ from both sides:
$13x=-25$
Divide both sides by $13$:
$x=-\dfrac{25}{13}$
So $x=-\dfrac{25}{13}$ is the only excluded value. The domain is all real numbers except this value.
In interval notation, the domain is
$\boxed{\left(-\infty,-\dfrac{25}{13}\right)\cup\left(-\dfrac{25}{13},\infty\right)}$
In set-builder notation, the domain is
$\boxed{\left\{x\mid x\text{ is a real number},\ x\ne -\dfrac{25}{13}\right\}}$
Example 4: Find the domain of the rational function $f(u)=\dfrac{7u-8}{u^2-64}$.
Solution:
Find the excluded values by setting the denominator equal to zero:
$u^2-64=0$
The left-hand side is a difference of squares, $u^2-8^2$, which factors as
$(u+8)(u-8)=0$
By the zero product rule, set each factor equal to zero:
$u+8=0\quad\text{or}\quad u-8=0$
Solving each gives
$u=-8\quad\text{or}\quad u=8$
So the excluded values are $-8$ and $8$. The domain is all real numbers except these two values.
In interval notation, the domain is
$\boxed{(-\infty,-8)\cup(-8,8)\cup(8,\infty)}$
In set-builder notation, the domain is
$\boxed{\{u\mid u\text{ is a real number},\ u\ne -8,\,8\}}$
Example 5: Find the domain of the rational function $f(x)=\dfrac{10x+13}{20}$.
Solution:
The denominator is the constant $20$. There is no variable in the denominator, so the denominator can never be zero, and there are no excluded values.
Therefore the domain is all real numbers.
In interval notation, the domain is
$\boxed{(-\infty,\infty)}$
In set-builder notation, the domain is
$\boxed{\{x\mid x\text{ is a real number}\}}$