Rational Function

What Is a Rational Function?

A rational function is a function that can be written as the ratio of two polynomial functions. Think of it like a fraction, but instead of having integers on the top and bottom, you have polynomials. The name "rational" comes from the word "ratio."

The standard form of a rational function is:

In this formula, both $P(x)$ (the numerator) and $Q(x)$ (the denominator) are polynomials. There's one very important rule: the denominator polynomial, $Q(x)$, cannot be equal to zero. Division by zero is undefined in mathematics, and this rule is the key to understanding many properties of rational functions.

Here are a few examples of rational functions:

• $f(x)=\frac{1}{x}$ (Here, $P(x)=1$ and $Q(x)=x$)
• $g(x)=\frac{x+2}{x-3}$ (Here, $P(x)=x+2$ and $Q(x)=x-3$)
• $h(x)=\frac{x^2-4}{x^2+5x+6}$ (Here, $P(x)=x^2-4$ and $Q(x)=x^2+5x+6$)

An expression is not a rational function if it contains things like square roots of variables or variables in an exponent. For example, $y=\frac{\sqrt{x}+1}{x-1}$ is not a rational function because its numerator contains a square root.

How Do You Find the Domain of a Rational Function?

The domain of a function is the set of all possible input values ($x$-values) for which the function is defined. For rational functions, the one and only rule you need to worry about is: the denominator cannot be zero. Therefore, to find the domain, you must find the $x$-values that would make the denominator zero and exclude them.

Here is the step-by-step process:

1. Take the polynomial in the denominator, $Q(x)$, and set it equal to zero.
2. Solve the equation $Q(x)=0$ for $x$.
3. The solutions you find are the values that are not in the domain. The domain is all real numbers except for these values.

What Are Vertical Asymptotes?

A vertical asymptote is an invisible vertical line that the graph of a function approaches but never touches or crosses. These asymptotes occur at the $x$-values where the function is undefined—the very same values we excluded from the domain!

Specifically, a vertical asymptote exists at $x=c$ if the denominator of the rational function is zero at $x=c$ and the numerator is not zero. When $x$ gets very close to $c$, the function's value shoots up to positive infinity or down to negative infinity.

To find the vertical asymptotes:

1. Simplify the rational function by factoring the numerator and denominator and cancelling any common factors. (We'll discuss what happens with cancelled factors later—they create 'holes' instead of asymptotes).
2. Set the simplified denominator equal to zero.
3. The solutions are the equations of your vertical asymptotes.

How Do You Find Horizontal Asymptotes?

A horizontal asymptote is an invisible horizontal line that the graph of a function approaches as $x$ heads towards positive infinity ($\mathrm{\infty }$) or negative infinity ($-\mathrm{\infty }$). It describes the long-term behavior or the 'ends' of the graph. A function can sometimes cross its horizontal asymptote, unlike a vertical one.

To find the horizontal asymptote of $f(x)=\frac{P(x)}{Q(x)}$, you only need to compare the degrees of the polynomials. Let the degree of the numerator $P(x)$ be $n$ and the degree of the denominator $Q(x)$ be $m$. There are three simple rules:

A Step-by-Step Guide to Graphing Rational Functions

Graphing a rational function involves putting all the pieces together: domain, asymptotes, and intercepts. Following a structured process makes it much easier.

Let's graph $f(x)=\frac{2x+2}{x-1}$.

1. Find Asymptotes:
   • Vertical Asymptote: Set the denominator to zero. $x-1=0⟹x=1$. The VA is at $x=1$.
   • Horizontal Asymptote: The degree of the numerator is 1, and the degree of the denominator is 1. Since the degrees are equal, we use the ratio of leading coefficients: $y=\frac{2}{1}=2$. The HA is at $y=2$.
2. Find Intercepts:
   • y-intercept: Set $x=0$. $f(0)=\frac{2(0)+2}{0-1}=\frac{2}{-1}=-2$. The y-intercept is at $(0,-2)$.
   • x-intercept(s): Set the numerator to zero. $2x+2=0⟹2x=-2⟹x=-1$. The x-intercept is at $(-1,0)$.
3. Plot Points and Sketch: Draw your asymptotes as dashed lines. Plot your intercepts. Now, test a point on each side of the vertical asymptote to see where the curve lies.
   • Left of $x=1$: Let's try $x=-2$. $f(-2)=\frac{2(-2)+2}{-2-1}=\frac{-2}{-3}=\frac{2}{3}$. So the point $(-2,2/3)$ is on the graph. This tells us the curve is above the horizontal asymptote on the far left.
   • Right of $x=1$: Let's try $x=3$. $f(3)=\frac{2(3)+2}{3-1}=\frac{8}{2}=4$. So the point $(3,4)$ is on the graph. This tells us the curve is above the horizontal asymptote on the right.
4. Draw the Graph: Connect the points with smooth curves that approach the asymptotes. The graph will have two separate branches, one on each side of the vertical asymptote.

What Is a 'Hole' in a Rational Function Graph?

Sometimes, a factor in the denominator cancels out with an identical factor in the numerator. When this happens, it doesn't create a vertical asymptote. Instead, it creates a hole in the graph, also known as a removable discontinuity. A hole is a single point where the function is undefined, which we show on the graph with a small open circle.

Consider the function $f(x)=\frac{x^2-4}{x-2}$. Let's find the hole.

1. Factor the numerator and denominator:
$$f(x)=\frac{(x-2)(x+2)}{x-2}$$
2. Identify and cancel the common factor: The factor $(x-2)$ is common to both. Cancelling it gives us a simplified function, let's call it $g(x)=x+2$.
3. Find the hole's location:
   • The $x$-coordinate of the hole is the value that made the cancelled factor zero. Set $x-2=0$, which gives $x=2$.
   • The $y$-coordinate is found by plugging this $x$-value into the simplified function. $g(2)=2+2=4$.

So, the graph of $f(x)=\frac{x^2-4}{x-2}$ looks exactly like the line $y=x+2$, but with a hole at the point $(2,4)$.

Common Mistakes to Avoid

• Illegal Cancellation: A very common error is to cancel terms instead of factors. For example, in $\frac{x+3}{x+5}$, you cannot cancel the $x$ terms to get $\frac{3}{5}$. Cancellation only works when the entire term is a factor of the top and bottom, like in $\frac{3(x+5)}{2(x+5)}$.
• Forgetting Negative Solutions: When solving a denominator like $x^2-16=0$, it's easy to find $x=4$ but forget the other solution, $x=-4$. This would cause you to miss a vertical asymptote.
• Confusing Holes and Asymptotes: Remember the rule: if a factor in the denominator cancels with a factor in the numerator, it creates a hole. If a factor in the denominator does not cancel, it creates a vertical asymptote.
• Mixing Up Horizontal Asymptote Rules: Keep the three rules for horizontal asymptotes straight. It's helpful to write them down (Bottom-Heavy, Equal Degrees, Top-Heavy) before you start a problem.
• Assuming a Graph Never Crosses a Horizontal Asymptote: While a graph never crosses its vertical asymptotes, it can and sometimes does cross its horizontal asymptote. The horizontal asymptote only dictates the behavior of the graph as $x$ approaches $\pm \mathrm{\infty }$.

Quick Summary and Key Concepts

Here is a quick reference table to help you remember how to find the key features of any rational function, $f(x)=\frac{P(x)}{Q(x)}$.