In Exercises 63-66, match the function with the corresponding graph by considering end behavior

step 1 65. We have x plus 2 over 2x squared plus 1. So what makes our denominator 0, I would have 2x sq

In Exercises 63-66, match the function with the...

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Key Concepts

End Behavior

The end behavior of a rational function describes how the function behaves as x grows large in magnitude. For this function, due to the degree of the numerator being less than the denominator, the function value approaches the horizontal asymptote y = 0 as x tends toward positive or negative infinity.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero while the numerator is nonzero. In this function, because the denominator 2x^2 + 1 is always positive and never zero, there are no vertical asymptotes.

Rational Functions

Rational functions are expressions formed by one polynomial divided by another. Analyzing such functions involves comparing the degrees of the numerator and denominator polynomials, which influences many properties, including asymptotic behavior and potential discontinuities.

Horizontal Asymptotes

When the degree of the numerator is less than the degree of the denominator, as is the case here, the horizontal asymptote of the rational function is y = 0. This indicates that as x approaches positive or negative infinity, the value of the function approaches zero.

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