Glossary
Denominator Zeros
The x-values for which the denominator polynomial q(x) of a rational function equals zero, indicating potential locations for vertical asymptotes or holes.
Example:
In the function g(x) = (x+5) / (x^2 - 9), the denominator zeros are x=3 and x=-3, which are critical points to analyze for asymptotes.
Hole (in a graph)
A point of discontinuity in the graph of a rational function where a common factor in the numerator and denominator cancels out, creating a 'gap' rather than an asymptote.
Example:
The function h(x) = (x^2 - 1) / (x - 1) has a hole at x=1 because the (x-1) factor cancels, making the graph identical to y=x+1 everywhere except at x=1.
Multiplicity (of a zero)
The number of times a specific factor (x-a) appears in the factored form of a polynomial, corresponding to how many times 'a' is a root.
Example:
In the polynomial P(x) = (x-1)^2 * (x+4)^3, the zero at x=1 has a multiplicity of 2, while the zero at x=-4 has a multiplicity of 3.
One-Sided Limits
The value a function approaches as the input variable approaches a specific point from either values less than that point (left-sided limit) or values greater than that point (right-sided limit).
Example:
When analyzing the behavior of f(x) = 1/(x-2) near its vertical asymptote, we use one-sided limits to see if the function approaches positive or negative infinity as x gets close to 2 from the left or right.
Rational Functions
Functions expressed as a ratio of two polynomial functions, typically in the form r(x) = p(x) / q(x), where q(x) is not the zero polynomial.
Example:
The cost per item for a company producing 'x' units might be modeled by a rational function like C(x) = (1000 + 5x) / x.
Vertical Asymptote
A vertical line that the graph of a function approaches infinitely closely but never touches, occurring where the function's value tends towards positive or negative infinity.
Example:
For the function f(x) = 1/(x-3), the line x=3 is a vertical asymptote, indicating where the function's output becomes unbounded.