(a) graph the rational function using transformations, (b)...

(a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes.
$$
R(x)=\frac{x^{2}-4}{x^{2}}
$$

Key Concepts

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. Understanding their properties, such as points of discontinuity, asymptotes, and intercepts, is crucial when analyzing their graphs. Their behavior is largely influenced by the degrees of the numerator and denominator polynomials.

Function Transformations

Transformations involve shifting, stretching, compressing, or reflecting the basic graph of a function to obtain the graph of a more complex function. With rational functions, understanding how to decompose a function into simpler parts via algebraic manipulation aids in graphing by revealing symmetry and asymptotic behavior.

Domain and Range

The domain of a function encompasses all allowable input values for which the function is defined, while the range is the collection of possible output values. For rational functions, the domain is influenced by restrictions that come from denominators being zero, and the range is often determined by analyzing horizontal or oblique asymptotes as well as other limits in function behavior.

Vertical Asymptotes

Vertical asymptotes are lines x = a where the rational function grows without bound as x approaches a value that makes the denominator zero (provided there is no cancellation with a factor in the numerator). They signal the presence of undefined points in the domain and indicate where the function's value diverges.

Horizontal and Oblique Asymptotes

Horizontal asymptotes describe the behavior of a function as x tends to infinity or negative infinity, typically determined by comparing the degrees of the numerator and denominator. Oblique asymptotes, on the other hand, occur when the degree of the numerator is exactly one more than that of the denominator. Both types of asymptotes help in understanding the end behavior of rational functions.

Transcript

Please give Ace some feedback