Rational Functions

What Are Rational Functions?

A rational function is any function that can be written as a fraction of two polynomials:

$$
f(x) = \frac{P(x)}{Q(x)},
$$

where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

Examples:

• $f(x) = \dfrac{2x+1}{x-3}$
• $g(x) = \dfrac{x^2 - 4}{x^2 + 1}$
• $h(x) = \dfrac{5}{x}$

Non-examples (not rational functions):

• $f(x) = \sqrt{x}$ (involves a root of $x$, not a ratio of polynomials)
• $g(x) = \sin x$ (trigonometric, not a ratio of polynomials)
• $h(x) = e^x$ (exponential, not a ratio of polynomials)

The key idea is: numerator and denominator are polynomials; the denominator is not identically zero.

Basic Features of Rational Functions

Although details like domain and asymptotes are treated in their own subsections, it helps to know the main geometric features you typically look for with rational functions:

• Values where the function is not defined (holes or vertical asymptotes).
• Long-term behavior as $x \to \infty$ or $x \to -\infty$ (horizontal or slant asymptotes).
• Zeros, where the numerator is zero but the denominator is not.

These features make rational functions behave differently from simple polynomial functions: their graphs often have breaks and approach lines without ever touching them.

Simplifying Rational Functions

Rational functions can often be simplified by factoring numerator and denominator and canceling common factors.

Example:

$$
f(x) = \frac{x^2 - 4}{x^2 - x - 6}.
$$

Factor:

• $x^2 - 4 = (x-2)(x+2)$
• $x^2 - x - 6 = (x-3)(x+2)$

So

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

For $x \neq -2$, you can cancel the common factor $(x+2)$:

$$
f(x) = \frac{x-2}{x-3}, \quad x \neq -2.
$$

Important: the simplified expression $\dfrac{x-2}{x-3}$ looks defined at $x=-2$, but the original function was not, because we divided by zero there. So:

• The formula simplifies.
• The function still has a missing point (a “hole”) at $x = -2$.

This is a typical feature of rational functions: cancellation changes the formula, but not where the function was originally undefined.

Zeros and Undefined Points

For a rational function
$$
f(x) = \frac{P(x)}{Q(x)},
$$
with factored form
$$
P(x) = a(x-r_1)(x-r_2)\cdots, \quad Q(x) = b(x-s_1)(x-s_2)\cdots,
$$
you can locate special $x$-values directly:

• A zero (or root) occurs at an $x$ where $P(x) = 0$ but $Q(x) \neq 0$.
• A value where the function is undefined occurs where $Q(x) = 0$.

Procedure:

1. Factor numerator and denominator fully.
2. Cancel any common factor (these give holes).
3. Remaining numerator zeros give $x$-intercepts.
4. Remaining denominator zeros give vertical asymptotes.

Example:

$$
f(x) = \frac{(x-1)(x+2)}{(x+2)(x-3)}.
$$

• Cancel $(x+2)$ (this creates a hole at $x=-2$).
• Simplified form: $f(x) = \dfrac{x-1}{x-3}$, but remember $x \neq -2$.

Then:

• Zero: $x=1$ (numerator zero, denominator nonzero).
• Undefined: $x=-2$ (hole, coming from canceled factor), and $x=3$ (vertical asymptote, factor not canceled).

Typical Shapes of Rational Function Graphs

While full graphing is detailed elsewhere, some patterns are specific to rational functions:

1. Simple reciprocal type: $f(x) = \dfrac{1}{x}$
• Graph has two branches, one in the first quadrant and one in the third.
• There is a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.
2. Shifted reciprocal: $f(x) = \dfrac{1}{x-2} + 3$
• Looks like $1/x$, but moved right by 2 and up by 3.
• Vertical asymptote at $x=2$; horizontal asymptote at $y=3$.
3. Higher-degree behavior: $f(x) = \dfrac{2x}{x^2+1}$
• Denominator grows faster as $|x|$ gets large, so the graph tends to a horizontal asymptote at $y=0$.
• Symmetry or skewness depends on the structure of $P(x)$ and $Q(x)$.

Recognizing these patterns helps when you sketch graphs or interpret rational functions in applications.

Operations with Rational Functions

You can add, subtract, multiply, and divide rational functions using the same fraction rules as for rational numbers, but with polynomials.

Let
$$
f(x) = \frac{P(x)}{Q(x)}, \quad g(x) = \frac{R(x)}{S(x)}.
$$

Assume denominators are not zero where expressions are evaluated.

Addition and Subtraction

Use common denominators:

• Addition:
  $$
  f(x) + g(x) = \frac{P(x)}{Q(x)} + \frac{R(x)}{S(x)}
  = \frac{P(x)S(x) + R(x)Q(x)}{Q(x)S(x)}.
  $$
• Subtraction:
  $$
  f(x) - g(x) = \frac{P(x)S(x) - R(x)Q(x)}{Q(x)S(x)}.
  $$

Simplify by factoring numerator and denominator and canceling common factors where possible (keeping track of domain restrictions).

Example:

$$
f(x) = \frac{1}{x}, \quad g(x) = \frac{2}{x+1}.
$$

Then

$$
f(x) + g(x) = \frac{1}{x} + \frac{2}{x+1}
= \frac{(1)(x+1) + (2)(x)}{x(x+1)}
= \frac{x+1+2x}{x(x+1)}
= \frac{3x+1}{x(x+1)}.
$$

The combined function has denominator $x(x+1)$, so it is undefined at $x=0$ and $x=-1$.

Multiplication

Multiply numerators and denominators:

$$
f(x) \cdot g(x) = \frac{P(x)}{Q(x)} \cdot \frac{R(x)}{S(x)}
= \frac{P(x)R(x)}{Q(x)S(x)}.
$$

Example:

$$
f(x) = \frac{x}{x-1}, \quad g(x) = \frac{x+2}{x+3}.
$$

Then

$$
f(x)g(x) = \frac{x(x+2)}{(x-1)(x+3)}.
$$

You can expand or factor further as needed, depending on the context.

Division

To divide by a nonzero rational function, multiply by its reciprocal:

$$
\frac{f(x)}{g(x)} = \frac{P(x)}{Q(x)} \div \frac{R(x)}{S(x)}
= \frac{P(x)}{Q(x)} \cdot \frac{S(x)}{R(x)}
= \frac{P(x)S(x)}{Q(x)R(x)},
$$

with the restriction that $R(x) \neq 0$ (so $g(x) \neq 0$) and $Q(x), S(x)$ are nonzero at the points you consider.

Example:

$$
f(x) = \frac{x^2}{x-4},\quad g(x) = \frac{x}{x+1}.
$$

Then

$$
\frac{f(x)}{g(x)} = \frac{x^2}{x-4} \div \frac{x}{x+1}
= \frac{x^2}{x-4} \cdot \frac{x+1}{x}
= \frac{x^2(x+1)}{x(x-4)}
= \frac{x(x+1)}{x-4},
$$

for $x \neq 0,4,-1$ (all original restrictions).

Simplifying Complex Rational Expressions

A complex rational expression is a fraction where the numerator or denominator (or both) are themselves rational expressions.

Example:

$$
F(x) = \frac{\dfrac{1}{x} + \dfrac{2}{x+1}}{\dfrac{3}{x}}.
$$

A standard technique is to clear denominators inside the big fraction:

1. Identify the least common denominator (LCD) of all small fractions.
2. Multiply numerator and denominator of the large fraction by that LCD.

In the example, the small denominators are $x$, $x+1$, and $x$. The LCD is $x(x+1)$.

Multiply top and bottom by $x(x+1)$:

• Numerator:
  $$
  x(x+1)\left(\frac{1}{x} + \frac{2}{x+1}\right)
  = (x+1) + 2x = 3x+1.
  $$
• Denominator:
  $$
  x(x+1)\cdot\frac{3}{x} = 3(x+1).
  $$

So

$$
F(x) = \frac{3x+1}{3(x+1)},
$$

with the original restrictions $x \neq 0,-1$.

This process converts a messy-looking expression into a single, simpler rational function.

Modeling with Rational Functions (Qualitative View)

Rational functions are often used to model situations involving:

• Rates (like speed, flow rate, work rate).
• Ratios (like concentration, density, or cost per item).
• Diminishing returns or saturation (quantity approaches a maximum limit).
• Inverse proportionality (when one quantity is proportional to $1/x$).

Examples of qualitative models:

• Average speed: total distance over total time, where both distance and time might depend on $x$, giving a ratio of two polynomials.
• Concentration: amount of substance divided by volume, both possibly polynomial functions of time.
• Cost per item: total cost divided by number of items, leading to expressions like
  $$
  C(x) = \frac{ax + b}{x},
  $$
  where $b$ might represent fixed costs and $ax$ variable cost.

The important idea here is recognizing when a relationship naturally takes the form “one polynomial quantity divided by another,” signaling that a rational function is an appropriate model, even before delving into full applications.