All Flashcards
Explain how to identify a hole in a rational function.
Factor the numerator and denominator. If a factor (x - a) is present in both, there's a hole at x = a.
How do you find the coordinates of a hole?
Find the x-value by setting the common factor to zero. Cancel the common factor, then evaluate the simplified function at that x-value to find the y-value.
Why does canceling the common factor 'repair' the function?
Canceling removes the division by zero at x = a, making the function defined everywhere except where other factors in the denominator are zero.
Explain the relationship between holes and limits.
Even though the function is undefined at the hole, the limit as x approaches the hole's x-value exists and is equal to the hole's y-value.
How does multiplicity affect the presence of a hole?
If the multiplicity of a zero in the numerator is greater than or equal to its multiplicity in the denominator, then there is a hole.
What happens if a factor is only in the denominator?
It creates a vertical asymptote, not a hole.
Why is factoring crucial when dealing with rational functions?
Factoring allows us to identify common factors, which indicate holes or simplifications that can be made.
Describe the graphical representation of a hole.
A hole appears as an open circle on the graph of the function at the point where the function is undefined.
Explain the difference between a hole and a vertical asymptote.
A hole is a removable discontinuity where the limit exists, while a vertical asymptote is a non-removable discontinuity where the limit is infinite or does not exist.
What does it mean for a function to have a 'removable discontinuity'?
It means the discontinuity (like a hole) can be 'removed' by redefining the function at that point, making it continuous.
What are the differences between a hole and a vertical asymptote?
Hole: Removable discontinuity, limit exists. | Vertical Asymptote: Non-removable discontinuity, limit is infinite or DNE.
Hole vs. Zero of a Function.
Hole: Factor cancels out. | Zero: Factor remains in numerator.
Removable vs. Non-Removable Discontinuity.
Removable: Can be 'fixed' by redefining the function. | Non-Removable: Cannot be 'fixed'; function approaches infinity or oscillates.
What are the differences between finding the x-coordinate of a hole and a vertical asymptote?
Hole: Set the common factor equal to zero. | Vertical Asymptote: Set the remaining factors in the denominator equal to zero.
Limit at a hole vs. Limit at a vertical asymptote.
Hole: Limit exists and is finite. | Vertical Asymptote: Limit is infinite or does not exist.
Rational function with a hole vs. Polynomial function.
Hole: Originally a rational function with common factors. | Polynomial: No division by a variable expression.
Hole vs. Point on a graph.
Hole: Function is undefined. | Point: Function has a defined value.
Graph of original function vs. Graph of simplified function (after removing hole).
Original: Hole is present. | Simplified: Hole is 'filled in'.
Factor in numerator only vs. Factor in both numerator and denominator.
Numerator Only: Zero of the function. | Both: Potential hole.
Hole vs. Jump Discontinuity
Hole: Limit exists. | Jump Discontinuity: Left and right limits exist but are not equal.
What is a hole in a rational function?
A point where the function is undefined because a factor is present in both the numerator and denominator.
Define 'multiplicity' of a zero.
The number of times a zero appears as a factor in the factorization of a polynomial.
What is a removable discontinuity?
A point on a function's graph that can be 'removed' by redefining the function at that point (e.g., a hole).
What is a rational function?
A function that can be defined as a quotient of two polynomials.
How does a hole relate to limits?
The y-coordinate of the hole is the limit of the function as x approaches the x-coordinate of the hole.
Define limit of a function.
The value that a function approaches as the input (or argument) approaches some value.
What does it mean for a function to be undefined at a point?
The function does not have a defined value at that specific input value, often due to division by zero.
What is a factor of a polynomial?
An expression that divides evenly into the polynomial, leaving no remainder.
What does it mean to 'cancel' a common factor?
To divide both the numerator and denominator of a fraction by the same factor, simplifying the expression.
What is the x-coordinate of a hole?
The value 'c' where the common factor (x-c) equals zero.