Rational Functions

If you have ever tried to split a pizza among friends or calculated how long a road trip takes at different speeds, you have already encountered rational relationships. When you divide one thing by another, the results can behave in surprising ways. Drive faster, and the time to your destination shrinks, but it never quite reaches zero no matter how fast you go. Add more people to share that pizza, and each person’s slice gets smaller and smaller, approaching nothing but never actually disappearing.

Rational functions capture this kind of behavior mathematically. They are the functions you get when you put one polynomial on top of another, like a fraction. This simple construction creates graphs that can split into multiple pieces, shoot off toward infinity, and approach invisible boundary lines called asymptotes. While that might sound complicated, you will find that rational functions follow predictable rules. Once you learn to spot the patterns, you can analyze and graph these functions with confidence.

Core Concepts

What Is a Rational Function?

A rational function is a function that can be written as a ratio of two polynomial functions:

$$f(x) = \frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

The word “rational” here comes from “ratio,” not from being reasonable or logical. You are simply taking one polynomial and dividing it by another.

Here are some examples of rational functions:

$$f(x) = \frac{1}{x} \qquad g(x) = \frac{x^2 - 4}{x - 3} \qquad h(x) = \frac{2x + 1}{x^2 + 5x + 6}$$

And here are some examples that are not rational functions:

$$f(x) = \frac{\sqrt{x}}{x + 1} \qquad g(x) = \frac{2^x}{x} \qquad h(x) = \frac{\sin x}{x}$$

These fail because the numerator or denominator contains something other than a polynomial (a square root, an exponential, a trigonometric function).

Domain Restrictions

The most important feature that distinguishes rational functions from polynomials is that rational functions have domain restrictions. Since division by zero is undefined, we must exclude any value of $x$ that makes the denominator equal to zero.

For example, consider:

$$f(x) = \frac{x + 3}{x - 2}$$

When $x = 2$, the denominator becomes $2 - 2 = 0$, which is undefined. So the domain of this function is all real numbers except 2, written as:

$$\text{Domain: } {x \in \mathbb{R} : x \neq 2}$$

or in interval notation:

$$(-\infty, 2) \cup (2, \infty)$$

To find domain restrictions, set the denominator equal to zero and solve for $x$. Those values must be excluded from the domain.

Vertical Asymptotes

When the denominator of a rational function equals zero but the numerator does not equal zero at that same value, the function has a vertical asymptote at that point.

A vertical asymptote is an invisible vertical line that the graph approaches but never touches or crosses. As $x$ approaches this value, the function values shoot off toward positive or negative infinity.

For $f(x) = \frac{x + 3}{x - 2}$:

• When $x = 2$: denominator = 0, numerator = $2 + 3 = 5 \neq 0$
• Therefore, there is a vertical asymptote at $x = 2$

The graph will approach the line $x = 2$ from both sides, going to $+\infty$ on one side and $-\infty$ on the other (or both going the same direction, depending on the function).

Holes (Removable Discontinuities)

Sometimes both the numerator and denominator equal zero at the same value of $x$. When this happens, the factor cancels out, and instead of a vertical asymptote, you get a hole in the graph.

Consider:

$$f(x) = \frac{x^2 - 9}{x - 3} = \frac{(x-3)(x+3)}{x-3}$$

When $x = 3$:

• Denominator = 0
• Numerator = $9 - 9 = 0$

Since both are zero, the factor $(x - 3)$ cancels:

$$f(x) = \frac{(x-3)(x+3)}{x-3} = x + 3 \quad \text{for } x \neq 3$$

The simplified function is just the line $y = x + 3$, but with a hole at $x = 3$. The function is undefined at $x = 3$, but there is no asymptote there. If you were to fill in the hole, its $y$-value would be $3 + 3 = 6$, so the hole is at the point $(3, 6)$.

Key distinction:

• If a factor in the denominator does not cancel with the numerator, it creates a vertical asymptote.
• If a factor in the denominator does cancel with a matching factor in the numerator, it creates a hole.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a rational function as $x$ approaches positive or negative infinity. It is a horizontal line that the graph approaches (but may cross) as you move far to the left or right.

The horizontal asymptote depends on comparing the degrees of the numerator and denominator:

Case 1: Degree of numerator < Degree of denominator

The horizontal asymptote is $y = 0$ (the x-axis).

Example: $f(x) = \frac{2x + 1}{x^2 + 3}$

Numerator degree: 1. Denominator degree: 2. Since $1 < 2$, the horizontal asymptote is $y = 0$.

Case 2: Degree of numerator = Degree of denominator

The horizontal asymptote is the ratio of the leading coefficients.

Example: $f(x) = \frac{3x^2 + 5x}{2x^2 - 1}$

Both have degree 2. Leading coefficients are 3 and 2. Horizontal asymptote: $y = \frac{3}{2}$.

Case 3: Degree of numerator > Degree of denominator

There is no horizontal asymptote. Instead, the function grows without bound (or may have an oblique asymptote, discussed next).

Oblique (Slant) Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique asymptote (also called a slant asymptote). This is a diagonal line that the graph approaches as $x \to \pm\infty$.

To find the oblique asymptote, perform polynomial long division. The quotient (ignoring the remainder) gives you the equation of the asymptote.

Example: $f(x) = \frac{x^2 + 2x + 3}{x - 1}$

Numerator degree: 2. Denominator degree: 1. Since $2 - 1 = 1$, there is an oblique asymptote.

Dividing $x^2 + 2x + 3$ by $x - 1$:

$$\frac{x^2 + 2x + 3}{x - 1} = x + 3 + \frac{6}{x - 1}$$

As $x \to \pm\infty$, the remainder $\frac{6}{x-1} \to 0$, so the graph approaches the line $y = x + 3$.

Finding Intercepts

X-intercepts occur where $f(x) = 0$. For a rational function $\frac{p(x)}{q(x)}$, this happens when the numerator $p(x) = 0$ (and the denominator is not also zero at that point).

Y-intercept is found by evaluating $f(0)$, assuming $x = 0$ is in the domain.

Notation and Terminology

Term	Symbol/Notation	Meaning
Rational function	$f(x) = \frac{p(x)}{q(x)}$	A ratio of two polynomials
Domain	${x : q(x) \neq 0}$	All $x$-values where the function is defined
Vertical asymptote	$x = a$	A vertical line the graph approaches but never crosses; occurs where denominator = 0 but numerator $\neq$ 0
Horizontal asymptote	$y = b$	A horizontal line the graph approaches as $x \to \pm\infty$
Oblique asymptote	$y = mx + b$	A slanted line the graph approaches as $x \to \pm\infty$; occurs when numerator degree is exactly one more than denominator degree
Hole	$(a, b)$	A point where the function is undefined due to a common factor that cancels
Removable discontinuity	Same as hole	The discontinuity can be “removed” by canceling common factors
Non-removable discontinuity	Vertical asymptote	The discontinuity cannot be removed; function is undefined and unbounded

Examples

Key Properties

Asymptote Summary

Condition	Asymptote Type	How to Find
Denominator = 0, numerator $\neq$ 0	Vertical asymptote	Solve $q(x) = 0$ after canceling common factors
Degree of $p(x)$ < Degree of $q(x)$	Horizontal at $y = 0$	N/A
Degree of $p(x)$ = Degree of $q(x)$	Horizontal at $y = \frac{a_n}{b_n}$	Ratio of leading coefficients
Degree of $p(x)$ = Degree of $q(x)$ + 1	Oblique asymptote	Perform polynomial division; quotient is the asymptote
Degree of $p(x)$ > Degree of $q(x)$ + 1	No linear asymptote	Function grows without bound

Behavior Near Vertical Asymptotes

Near a vertical asymptote at $x = a$:

• The function approaches $+\infty$ or $-\infty$ from each side
• The graph never crosses a vertical asymptote
• To determine which direction the graph goes, test values close to $a$ from both sides

Behavior Near Horizontal Asymptotes

Unlike vertical asymptotes:

• The graph can cross a horizontal asymptote (just not as $x \to \pm\infty$)
• The graph approaches the horizontal asymptote as $x$ becomes very large or very small
• To check if the graph crosses the horizontal asymptote, solve $f(x) = b$ where $y = b$ is the asymptote

Symmetry

Some rational functions have symmetry:

Even function (symmetric about y-axis): $f(-x) = f(x)$

Example: $f(x) = \frac{x^2}{x^2 - 1}$

Odd function (symmetric about origin): $f(-x) = -f(x)$

Example: $f(x) = \frac{x}{x^2 - 1}$

Real-World Applications

Average Cost

In business, if the fixed cost to set up production is $C_0$ and the variable cost per unit is $v$, then the average cost per unit when producing $x$ units is:

$$A(x) = \frac{C_0 + vx}{x} = \frac{C_0}{x} + v$$

As production increases ($x \to \infty$), the average cost approaches $v$, the variable cost per unit. This is why mass production reduces per-unit costs: the fixed costs get spread over more units.

Concentration of Medicine

When medicine is administered, its concentration in the bloodstream over time often follows a rational function:

$$C(t) = \frac{at}{t^2 + b}$$

The concentration rises, peaks, then decreases as the body metabolizes the drug. The horizontal asymptote at $C = 0$ means the drug eventually leaves the system.

Electrical Circuits

In parallel circuits, the total resistance $R$ from two resistors $R_1$ and $R_2$ is given by:

$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$

Solving for $R$:

$$R = \frac{R_1 R_2}{R_1 + R_2}$$

This is a rational function of either resistance.

Speed, Time, and Distance

The relationship $\text{time} = \frac{\text{distance}}{\text{speed}}$ is a rational function. If you drive 300 miles:

$$t(s) = \frac{300}{s}$$

The vertical asymptote at $s = 0$ reflects that you cannot arrive if you do not move. The horizontal asymptote at $t = 0$ shows that even infinite speed cannot make the time actually zero.

Optics and Lenses

The thin lens equation relates object distance $d_o$, image distance $d_i$, and focal length $f$:

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$

Solving for any one variable in terms of another gives a rational function with important asymptotes that describe where images form and how they change.

Self-Test Problems

Summary

• A rational function has the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

• The domain excludes all values where the denominator equals zero.

• Vertical asymptotes occur at values where the denominator equals zero but the numerator does not. The graph approaches infinity near these lines.

• Holes (removable discontinuities) occur when a factor cancels from both numerator and denominator. The function is undefined at that point, but the graph has a single missing point rather than an asymptote.

• Horizontal asymptotes depend on the degrees of numerator and denominator:

  • Numerator degree < denominator degree: $y = 0$
  • Numerator degree = denominator degree: $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
  • Numerator degree > denominator degree: no horizontal asymptote

• Oblique (slant) asymptotes occur when the numerator degree is exactly one more than the denominator degree. Find it by polynomial long division.

• X-intercepts occur where the numerator equals zero (provided the denominator is not also zero there).

• Y-intercept is found by evaluating $f(0)$, if 0 is in the domain.

• To solve rational equations, multiply by the LCD to clear denominators, solve the resulting equation, then check that solutions do not make any original denominator zero (extraneous solutions).

• Rational functions model many real-world situations including average costs, drug concentrations, electrical circuits, and relationships between speed and time.