Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In the given function ƒ(x)=(x+7)/(x+1), the numerator is a polynomial of degree one, and the denominator is also a polynomial of degree one. Understanding the structure of rational functions is essential for analyzing their behavior, including asymptotes and intercepts.
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Intro to Rational Functions
Vertical Asymptotes
Vertical asymptotes occur in rational functions where the denominator equals zero, leading to undefined values. For the function ƒ(x)=(x+7)/(x+1), setting the denominator x+1 to zero reveals a vertical asymptote at x=-1. This concept is crucial for understanding the limits and behavior of the function near these points.
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Determining Vertical Asymptotes
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a rational function as x approaches infinity or negative infinity. For the function ƒ(x)=(x+7)/(x+1), the degrees of the numerator and denominator are the same, indicating a horizontal asymptote at y=1, which is found by taking the ratio of the leading coefficients. This concept helps in predicting the long-term behavior of the function.
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Determining Horizontal Asymptotes