AP PreCalculus Flashcards: Rational Functions and Zeros
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is a 'zero' of a rational function?
A zero of a rational function is an x-value within its domain that makes the function's output equal to zero.
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What is a 'zero' of a rational function?
A zero of a rational function is an x-value within its domain that makes the function's output equal to zero.
How are the real zeros of a rational function determined?
The real zeros of a rational function correspond to the real zeros of its numerator, for any values that are within the function's domain.
What are the real zeros of the rational function r(x) = (x-2)(x+5) / (x-1)?
The real zeros are the zeros of the numerator, which are x=2 and x=-5, as both are in the function's domain.
Why do the zeros of a rational function only correspond to the zeros of the numerator?
A fraction is equal to zero only when its numerator is zero, provided the denominator is not also zero (which would make the expression undefined).
What two roles can the real zeros of a rational function's denominator play when solving an inequality?
The real zeros of the denominator act as endpoints for test intervals and correspond to the locations of vertical asymptotes (or holes) for the function.
Does the function r(x) = (x-4) / (x-4) have a zero at x=4?
No, because x=4 is a zero of the numerator but is not in the domain of r(x), as it also makes the denominator zero.
How do the real zeros of a rational function relate to its graph?
The real zeros of a rational function correspond to the x-intercepts of its graph.
To solve the inequality (x+3)/(x-7) ≤ 0, what critical values must be identified first?
The critical values are the real zeros of the numerator (x=-3) and the denominator (x=7), which will be used as interval endpoints.
Explain the significance of the domain when finding the zeros of a rational function.
A value that is a zero of the numerator cannot be a zero of the rational function if it is not in the domain (i.e., it also makes the denominator zero).
When solving a rational inequality like r(x) ≥ 0, what values serve as endpoints for the test intervals?
The real zeros of both the numerator and the denominator of the rational function serve as the endpoints for the intervals.