Mastering Rational Functions: Properties, Graphing, And Simplification Techniques

Rational Function

f(x) = 1/x

A rational function is a type of mathematical function that ratios two polynomial functions. It takes the general form of f(x) = P(x)/Q(x), where P(x) and Q(x) are both polynomial functions with real coefficients and Q(x) is not equal to zero. The domain of a rational function consists of all the values of x that make the denominator Q(x) not equal to zero.

Some properties of rational functions include:

1. Asymptotes: Rational functions have vertical and/or horizontal asymptotes, which are lines that the graph of the function approaches but never touches.

2. Zeros: The zeros of a rational function are the values of x that make the numerator P(x) equal to zero.

3. Domain and Range: The domain of a rational function consists of all the values of x that make the denominator Q(x) not equal to zero, while the range can be found by analyzing the asymptotes and zeros of the function.

4. Graphing: Graphing a rational function involves plotting the zeros, asymptotes, and any intercepts of the function and analyzing the behavior of the function in between each of these points.

5. Simplification: Rational functions can often be simplified by factoring the numerator and denominator and canceling any common factors.

Overall, rational functions are an important class of functions in mathematics, and they have many applications in various fields such as engineering, physics, and economics.

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