What are the equations of the asymptotes of the graph of the function [tex]f(x)=\frac{3 x^2-2 x-1}{x^2+3 x-10}[/tex]?
A. [tex]x=-5, x=2[/tex] and [tex]y=3[/tex]
B. [tex]x=-2, x=5[/tex] and [tex]y=3[/tex]
C. [tex]x=3, y=-5[/tex], and [tex]y=2[/tex]
D. [tex]x=3, y=-2[/tex], and [tex]y=5[/tex]
Answer (2)
Find the vertical asymptotes by setting the denominator equal to zero: x 2 + 3 x − 10 = 0 , which factors to ( x + 5 ) ( x − 2 ) = 0 , giving x = − 5 and x = 2 .
Verify that the numerator is non-zero at these points: 3 ( − 5 ) 2 − 2 ( − 5 ) − 1 = 84 = 0 and 3 ( 2 ) 2 − 2 ( 2 ) − 1 = 7 = 0 .
Find the horizontal asymptote by taking the limit as x approaches infinity: lim x → ∞ x 2 + 3 x − 10 3 x 2 − 2 x − 1 = 3 .
The equations of the asymptotes are x = − 5 , x = 2 , and y = 3 , so the answer is x = − 5 , x = 2 and y = 3 .
Explanation
Problem Analysis We are given the function f ( x ) = x 2 + 3 x − 10 3 x 2 − 2 x − 1 and we need to find the equations of its asymptotes.
Asymptote Properties Asymptotes can be vertical or horizontal. Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Horizontal asymptotes occur when x approaches infinity. The horizontal asymptote is the ratio of the leading coefficients when the degree of the numerator and denominator are the same.
Finding Vertical Asymptotes To find the vertical asymptotes, we need to find the values of x for which the denominator x 2 + 3 x − 10 is equal to zero. So, we solve the equation x 2 + 3 x − 10 = 0 .
Solving for Vertical Asymptotes We can factor the quadratic equation as ( x + 5 ) ( x − 2 ) = 0 . Thus, the roots are x = − 5 and x = 2 .
Checking Numerator Now we need to check if the numerator is non-zero at these roots. The numerator is 3 x 2 − 2 x − 1 . Let's evaluate the numerator at x = − 5 and x = 2 .
Verifying Vertical Asymptotes At x = − 5 , the numerator is 3 ( − 5 ) 2 − 2 ( − 5 ) − 1 = 3 ( 25 ) + 10 − 1 = 75 + 10 − 1 = 84 = 0 . At x = 2 , the numerator is 3 ( 2 ) 2 − 2 ( 2 ) − 1 = 3 ( 4 ) − 4 − 1 = 12 − 4 − 1 = 7 = 0 . Since the numerator is non-zero at both x = − 5 and x = 2 , we have vertical asymptotes at x = − 5 and x = 2 .
Finding Horizontal Asymptote To find the horizontal asymptote, we need to find the limit of the function as x approaches infinity: lim x → ∞ x 2 + 3 x − 10 3 x 2 − 2 x − 1 . Since the degree of the numerator and denominator are the same (both are 2), the horizontal asymptote is the ratio of the leading coefficients, which is 1 3 = 3 . Thus, the horizontal asymptote is y = 3 .
Final Answer Therefore, the equations of the asymptotes are x = − 5 , x = 2 , and y = 3 .
Examples
Asymptotes are useful in various real-world applications. For example, in physics, the velocity of an object approaching the speed of light has an asymptote at the speed of light. In economics, cost functions may have asymptotes representing maximum production capacity. In population modeling, asymptotes can represent the carrying capacity of an environment, which is the maximum sustainable population size. Understanding asymptotes helps us analyze and predict the behavior of functions in extreme conditions.
The equations of the asymptotes for the function f ( x ) = x 2 + 3 x − 10 3 x 2 − 2 x − 1 are vertical asymptotes at x = − 5 and x = 2 and a horizontal asymptote at y = 3 . Therefore, the correct answer is option A: x = − 5 , x = 2 and y = 3 .
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