3.7 Rational Functions

Rational Function Notation and Domain

Rational functions are ratios of two polynomials, written as $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$. They show up constantly when modeling rates, proportions, and inverse relationships. The key challenge with these functions is that division by zero creates breaks in the graph, so much of your work will focus on finding where those breaks happen and what the function does near them.

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Arrow notation for rational functions

Arrow notation expresses a rational function as $f: X \to Y$, where $X$ is the domain (all valid inputs) and $Y$ is the codomain (possible outputs). This notation is a compact way to communicate what goes in and what comes out.

For rational functions, the domain includes all real numbers except values that make the denominator equal zero. Those excluded values are where the interesting behavior happens: vertical asymptotes or holes.

Domain restrictions of rational functions

To find the domain, set the denominator equal to zero and solve. Whatever x-values you get are excluded.

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

1. Set the denominator equal to zero: $x - 2 = 0$

2. Solve: $x = 2$

3. Domain is all real numbers except $x = 2$, written as $\{x \in \mathbb{R} \mid x \neq 2\}$

For denominators that are higher-degree polynomials, you may need to factor first. For instance, if the denominator is $x^2 - 9$, factor it as $(x-3)(x+3)$ and exclude both $x = 3$ and $x = -3$.

Rational Function Applications and Graphing

Real-world applications of rational functions

Rational functions model situations where one quantity depends on another through division:

• Rates: Average speed is $\frac{\text{distance}}{\text{time}}$, and concentration is $\frac{\text{amount of solute}}{\text{volume of solution}}$
• Inverse variations: If $y = \frac{k}{x}$, doubling $x$ halves $y$. This applies to scenarios like splitting a fixed cost among more people.
• Proportions: Scale factors and similarity ratios often involve rational expressions.

To solve a real-world problem with rational functions:

1. Identify the given quantities and what you need to find.
2. Write a rational function that captures the relationship.
3. Solve the equation or evaluate the function at the relevant input.

Graphing techniques for rational functions

Follow these steps to graph a rational function:

1. Find the domain by setting the denominator equal to zero. These x-values are where vertical asymptotes or holes occur.
2. Check for holes by factoring numerator and denominator. If a factor cancels, that x-value is a hole, not a vertical asymptote.
3. Find intercepts. Set $x = 0$ for the y-intercept. Set the numerator equal to zero for x-intercepts (but only values that are actually in the domain).
4. Determine the horizontal or slant asymptote using the degree comparison rules below.
5. Plot a few extra points on each side of every vertical asymptote, then sketch the curve.

Horizontal asymptote rules

Compare the degree of the numerator ($n$) to the degree of the denominator ($d$):

• $n < d$: Horizontal asymptote at $y = 0$
• $n = d$: Horizontal asymptote at $y = \frac{a_n}{b_d}$, where $a_n$ and $b_d$ are the leading coefficients of the numerator and denominator
• $n = d + 1$: No horizontal asymptote. Instead, there's a slant (oblique) asymptote, found by performing polynomial long division.
• $n > d + 1$: No horizontal asymptote. The end behavior resembles a polynomial (the quotient from long division).

Rational Function Asymptotes and Behavior

Asymptotes in rational functions

Vertical asymptotes occur at x-values where the denominator equals zero and the factor does not cancel with the numerator. As $x$ approaches a vertical asymptote, the function shoots toward $+\infty$ or $-\infty$.

Horizontal asymptotes describe what happens to $f(x)$ as $x \to \pm\infty$. The function gets closer and closer to the asymptote value but (typically) never reaches it at the extremes.

Behavior analysis of rational functions

Near a vertical asymptote, check the sign of the function on both sides. Plug in a value just to the left and just to the right of the asymptote to determine whether the function goes to $+\infty$ or $-\infty$ on each side. The sign doesn't always change; it depends on the multiplicity of the factor in the denominator.

• Odd multiplicity in the denominator (e.g., $(x-2)^1$): the function goes in opposite directions on either side of the asymptote.
• Even multiplicity (e.g., $(x-2)^2$): the function goes in the same direction on both sides.

Near a horizontal asymptote, the function approaches the asymptote value as $x \to \pm\infty$. It may approach from above on one side and below on the other, or from the same side on both.

Discontinuities and Special Cases

Types of discontinuities in rational functions

• Vertical asymptotes (non-removable): The denominator equals zero but the numerator does not at that x-value. The function is undefined and the graph has a break that can't be "fixed."
• Holes (removable discontinuities): Both the numerator and denominator equal zero at the same x-value, meaning a common factor cancels out. The graph has a single missing point. To find the y-coordinate of the hole, cancel the common factor and evaluate the simplified function at that x-value.
• Jump discontinuities: These only occur in piecewise-defined rational functions, not in a single rational expression.

Example of a hole: For $f(x) = \frac{(x-3)(x+1)}{(x-3)(x+2)}$, the factor $(x-3)$ cancels. There's a hole at $x = 3$. The simplified function is $\frac{x+1}{x+2}$, so the hole is at $\left(3,\, \frac{4}{5}\right)$. There's still a vertical asymptote at $x = -2$.

Special rational functions

• Reciprocal functions have a constant numerator and a variable denominator, like $f(x) = \frac{1}{x}$ or $f(x) = \frac{1}{x^2}$. These are the parent functions for many rational graphs. The graph of $\frac{1}{x}$ has vertical and horizontal asymptotes at $x = 0$ and $y = 0$, with branches in quadrants I and III.
• Polynomial functions are technically rational functions where the denominator is a nonzero constant (e.g., $\frac{x^2 + 3x}{1}$). Since the denominator never equals zero, their graphs are continuous with no asymptotes or holes.