Rational Functions | General Mathematics
Updated: November 18, 2024
Summary
The video explains rational functions as functions in the form r(x) = p(x)/q(x), where q(x) is not equal to zero and p(x) and q(x) are polynomials. It provides examples like r(x) = 1/x^2 and r(x) = (4x+4)/(x-2) to illustrate rational functions. The video discusses how to identify rational functions by examining the form of the function and the polynomials involved, such as f(x) = x^2 + 3 / x + 2 and f(x) = (x^3 + 2x - 1) / (2x - 1). It also explains a function that is not a rational function, like f(x) = sqrt(x^2 + 1) / (x - 5), due to the presence of a radical in the function.
Definition of Rational Function
Defines a rational function as a function of the form r(x) = p(x)/q(x) where q(x) is not equal to zero and p(x) and q(x) are polynomials.
Examples of Rational Functions
Provides examples of rational functions such as r(x) = 1/x^2, r(x) = (4x+4)/(x-2), and others showing how they fit the definition of rational functions.
Identifying Rational Functions
Discusses how to identify rational functions by looking at the form of the function and the polynomials involved, with examples like f(x) = x^2 + 3 / x + 2 and f(x) = (x^3 + 2x - 1) / (2x - 1).
Non-Rational Functions
Explains a function that is not a rational function like f(x) = sqrt(x^2 + 1) / (x - 5) due to the presence of a radical in the function.
FAQ
Q: What is a rational function?
A: A rational function is a function of the form r(x) = p(x)/q(x) where q(x) is not equal to zero and both p(x) and q(x) are polynomials.
Q: Can you provide examples of rational functions?
A: Examples of rational functions include r(x) = 1/x^2 and r(x) = (4x+4)/(x-2), where the function is defined as a ratio of two polynomials.
Q: How can you identify a rational function?
A: Rational functions can be identified by looking at the form of the function, which is a ratio of two polynomials, and ensuring that the denominator polynomial is not zero.
Q: Could you explain why f(x) = sqrt(x^2 + 1) / (x - 5) is not a rational function?
A: The function f(x) = sqrt(x^2 + 1) / (x - 5) is not a rational function because it contains a radical term within the function, which is not in the form of a ratio of two polynomials.
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