Glossary

Critical Points

Criticality: 3

The x-values that divide the number line into intervals where a function's sign might change. For rational functions, these are the real zeros of both the numerator and the denominator.

Example:

For the function $f(x) = \frac{x(x-4)}{(x+1)(x-2)}$ , the critical points are x = -1, x = 0, x = 2, and x = 4.

Denominator

Criticality: 2

The polynomial expression located below the fraction bar in a rational function.

Example:

For the function $k(x) = \frac{x^2 + 4}{x - 7}$ , the denominator is $x - 7$ .

Domain

Criticality: 3

The set of all possible input values (x-values) for which a function is defined. For rational functions, it excludes any x-values that make the denominator zero.

Example:

The domain of the function $f(x) = \frac{1}{x-4}$ is all real numbers except x = 4, because the function is undefined at that point.

Interval Analysis

Criticality: 3

A method used to determine the sign (positive or negative) of a function over different intervals on the x-axis, typically separated by zeros and vertical asymptotes.

Example:

To solve the inequality $\frac{x-1}{x+2} > 0$ , you would use interval analysis by testing values in the regions defined by x = 1 and x = -2.

Numerator

Criticality: 2

The polynomial expression located above the fraction bar in a rational function.

Example:

In the rational function $h(x) = \frac{x^2 + 2x - 3}{x - 1}$ , the numerator is $x^2 + 2x - 3$ .

Rational Functions

Criticality: 3

A function that can be expressed as the ratio of two polynomials, where the denominator polynomial is not identically zero.

Example:

The function $f(x) = \frac{x^2 - 1}{x + 3}$ is a rational function because it is a polynomial divided by another polynomial.

Real Zeros

Criticality: 3

The x-values where a function's output is zero, meaning the graph intersects the x-axis. For rational functions, these are the real zeros of the numerator that are also within the function's domain.

Example:

For the function $g(x) = \frac{x-5}{x+2}$ , the real zero is x = 5, since $g(5) = 0$ .

Vertical Asymptotes

Criticality: 3

Vertical lines that the graph of a rational function approaches but never touches. They occur at x-values where the denominator is zero and the numerator is non-zero.

Example:

The function $y = \frac{x+1}{x-3}$ has a vertical asymptote at x = 3, indicating the graph will get infinitely close to this line.