In Problems 36-38, find all horizontal and vertical asymptotes for each rational function. f(x)=

step 1 Okay, so let's find the horizontal and vertical asymptotes of this given f -of -x function, okay

In Problems 36-38, find all horizontal and vertical...

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Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined due to the denominator equating to zero, provided that there is no common factor with the numerator that would cancel out the discontinuity. They represent values where the function's output grows without bound, and they are critical in understanding the domain and discontinuities of a rational function.

Rational Functions

Rational functions are functions that are expressed as the ratio of two polynomials. Their behavior, particularly the existence of asymptotes, depends on the degrees of these polynomials as well as the coefficients of the leading terms. It is important to study rational functions to understand how they behave at extreme values and where they might be undefined.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the independent variable approaches positive or negative infinity. For rational functions, if the degree of the numerator equals the degree of the denominator, the horizontal asymptote is determined by the ratio of the leading coefficients. When the degree of the numerator is lower than that of the denominator, the horizontal asymptote is y=0.

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