Essential Concepts
Rational Functions
Definition
A rational function is a function that can be written as the quotient of two polynomials:
[latex]f(x) = \dfrac{P(x)}{Q(x)} = \dfrac{a_px^p + a_{p-1}x^{p-1} + \cdots + a_1x + a_0}{b_qx^q + b_{q-1}x^{q-1} + \cdots + b_1x + b_0}[/latex]
where [latex]Q(x) \ne 0[/latex]
Arrow Notation
Arrow notation describes the behavior of a function as inputs or outputs approach specific values:
| Symbol | Meaning |
| [latex]x \to a^-[/latex] | [latex]x[/latex] approaches [latex]a[/latex] from the left ([latex]x < a[/latex] but close to [latex]a[/latex]) |
| [latex]x \to a^+[/latex] | [latex]x[/latex] approaches [latex]a[/latex] from the right ([latex]x > a[/latex] but close to [latex]a[/latex]) |
| [latex]x \to \infty[/latex] | [latex]x[/latex] increases without bound |
| [latex]x \to -\infty[/latex] | [latex]x[/latex] decreases without bound |
| [latex]f(x) \to \infty[/latex] | output increases without bound |
| [latex]f(x) \to -\infty[/latex] | output decreases without bound |
| [latex]f(x) \to a[/latex] | output approaches [latex]a[/latex] |
Domain of Rational Functions
The domain includes all real numbers except those that cause the denominator to equal zero.
To find the domain:
- Set the denominator equal to zero
- Solve for [latex]x[/latex]
- Exclude these values from the domain
Vertical Asymptotes
A vertical asymptote is a vertical line [latex]x = a[/latex] where the graph tends toward positive or negative infinity as inputs approach [latex]a[/latex].
To find vertical asymptotes:
- Factor the numerator and denominator
- Cancel common factors
- Set the remaining denominator equal to zero and solve
- These values are vertical asymptotes
Removable Discontinuities (Holes)
A removable discontinuity occurs when a factor appears in both the numerator and denominator.
To find holes:
- Factor numerator and denominator completely
- Identify common factors
- Set the common factor equal to zero and solve
- This [latex]x[/latex]-value is the location of the hole
Horizontal Asymptotes
A horizontal asymptote is a horizontal line [latex]y = b[/latex] that the graph approaches as [latex]x \to \pm\infty[/latex].
Compare degrees of numerator and denominator:
Case 1: Degree of numerator < degree of denominator
-
- Horizontal asymptote: [latex]y = 0[/latex]
Case 2: Degree of numerator = degree of denominator
-
- Horizontal asymptote: [latex]y = \frac{a_n}{b_n}[/latex] (ratio of leading coefficients)
Case 3: Degree of numerator > degree of denominator by 1
-
- No horizontal asymptote; has a slant asymptote instead
Slant (Oblique) Asymptotes
A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.
To find the slant asymptote:
- Divide the numerator by the denominator using polynomial long division
- The quotient (ignoring the remainder) is the equation of the slant asymptote
Intercepts of Rational Functions
x-intercepts: Set the numerator equal to zero and solve (as long as the denominator is not also zero at those points)
y-intercept: Evaluate [latex]f(0)[/latex], if defined
Graphing Rational Functions
Steps to graph:
- Simplify by factoring and canceling common factors
- Find the domain (exclude values where denominator = 0)
- Identify vertical asymptotes and holes
- Find horizontal or slant asymptote
- Find intercepts (x-intercepts and y-intercept)
- Plot key points near asymptotes if needed
- Sketch the graph using asymptotes and intercepts
Writing Rational Functions from Graphs
If a rational function has:
- x-intercepts at [latex]x_1, x_2, \ldots, x_n[/latex]
- vertical asymptotes at [latex]v_1, v_2, \ldots, v_m[/latex]
Then: [latex]f(x) = a\dfrac{(x - x_1)^{p_1}(x - x_2)^{p_2} \cdots (x - x_n)^{p_n}}{(x - v_1)^{q_1}(x - v_2)^{q_2} \cdots (x - v_m)^{q_m}}[/latex]
where [latex]p_i[/latex] and [latex]q_i[/latex] are determined by graph behavior, and [latex]a[/latex] is found using a point on the graph.
Variation
Direct Variation
[latex]y[/latex] varies directly with [latex]x^n[/latex] if:
[latex]y = kx^n[/latex]
where [latex]k[/latex] is the constant of variation (nonzero).
Inverse Variation
[latex]y[/latex] varies inversely with [latex]x^n[/latex] if:
[latex]y = \dfrac{k}{x^n}[/latex]
where [latex]k[/latex] is the constant of variation (nonzero).
Joint Variation
Joint variation occurs when a variable depends on multiple other variables.
Examples:
- [latex]x[/latex] varies directly with [latex]y[/latex] and [latex]z[/latex]: [latex]x = kyz[/latex]
- [latex]x[/latex] varies directly with [latex]y[/latex] and inversely with [latex]z[/latex]: [latex]x = \frac{ky}{z}[/latex]
Rational Equations and Inequalities
Solving Rational Equations
A rational equation contains at least one rational expression with a variable in the denominator.
Steps:
- Factor all denominators
- Find and exclude values that make denominators zero
- Find the LCD (Least Common Denominator)
- Multiply the entire equation by the LCD
- Solve the resulting equation
- Check solutions in the original equation
Important: Always check that solutions don’t make any denominator zero. These must be excluded.
Solving Rational Inequalities
A rational inequality contains a rational expression with an inequality symbol.
Steps:
- Write the inequality with zero on one side
- Identify critical values:
- Zeros of numerator
- Zeros of denominator (excluded values)
- Create a sign chart testing each interval
- Determine which intervals satisfy the inequality
- Write the solution using interval notation
Key Equations
| Rational function | [latex]f(x) = \dfrac{P(x)}{Q(x)}[/latex], where [latex]Q(x) \ne 0[/latex] |
| Direct variation | [latex]y = kx^n[/latex], [latex]k[/latex] is a nonzero constant |
| Inverse variation | [latex]y = \dfrac{k}{x^n}[/latex], [latex]k[/latex] is a nonzero constant |
Glossary
arrow notation
a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value
constant of variation
the nonzero value [latex]k[/latex] that helps define the relationship between variables in direct or inverse variation
direct variation
a relationship where one quantity is a constant multiplied by another quantity; as one quantity increases, so does the other
horizontal asymptote
a horizontal line [latex]y = b[/latex] where the graph approaches the line as the inputs increase or decrease without bound
inverse variation
a relationship where one quantity is a constant divided by another quantity; as one quantity increases, the other decreases
inversely proportional
a relationship where one quantity is a constant divided by the other quantity
joint variation
a relationship where a variable varies directly or inversely with multiple variables
rational equation
an equation that contains at least one rational expression with a variable in the denominator
rational function
a function that can be written as the ratio of two polynomials
rational inequality
an inequality that contains at least one rational expression with a variable in the denominator
removable discontinuity
a single point at which a function is undefined that, if filled in, would make the function continuous; appears as a hole on the graph
varies directly
a relationship where one quantity is a constant multiplied by another quantity
varies inversely
a relationship where one quantity is a constant divided by another quantity
vertical asymptote
a vertical line [latex]x = a[/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[/latex]