Glossary
Asymptote
A line that a curve approaches as it heads towards infinity, but never actually reaches or crosses.
Example:
The function f(x) = 1/x has a vertical asymptote at x=0 and a horizontal asymptote at y=0.
Denominator
The bottom part of a fraction or rational expression, indicating the divisor.
Example:
In the rational function f(x) = (x+5)/(x-1), (x-1) is the denominator.
Factor
To express a polynomial as a product of simpler polynomials (verb); a term that divides another term exactly (noun).
Example:
To find holes, you must first factor the numerator and denominator, where (x-a) would be a common factor.
Finite Limit
A specific, real number that a function approaches as its input approaches a certain value.
Example:
As x approaches 0, the function f(x) = x² approaches a finite limit of 0.
Hole
A point where a rational function is undefined due to a common factor in both the numerator and denominator, but where the function can be 'repaired' by canceling the common factor.
Example:
For the function f(x) = (x² - 4)/(x - 2), there is a hole at x=2 because (x-2) is a common factor that cancels out.
Multiplicity
The number of times a particular zero appears as a root in the factorization of a polynomial.
Example:
In the polynomial (x-5)³(x+1)², the zero x=5 has a multiplicity of 3.
Numerator
The top part of a fraction or rational expression, representing the dividend.
Example:
In the rational function f(x) = (x+5)/(x-1), (x+5) is the numerator.
Rational Function
A function that can be expressed as the ratio of two polynomial functions, where the denominator is not equal to zero.
Example:
The function r(x) = (x² + 2x) / (x - 3) is an example of a rational function.
Removable Discontinuity
A type of discontinuity in a function where the function can be made continuous by redefining it at a single point; a hole is an example of this.
Example:
The graph of f(x) = (x² - 9)/(x - 3) has a removable discontinuity at x=3, which appears as a hole.