Select the correct answer.
Which statement describes the graph of the function [tex]f(x)=\frac{x^2-1}{x^2-2 x+1}[/tex]?
A. There is a hole at [tex]x=-1[/tex].
B. There is a vertical asymptote at [tex]x=-1[/tex].
C. The [tex]y[/tex]-intercept is [tex]y=-1[/tex].
D. There is a horizontal asymptote at [tex]y=-1[/tex].
Answer (2)
Factor the numerator and denominator of the function: f ( x ) = ( x − 1 ) 2 ( x − 1 ) ( x + 1 ) .
Simplify the function: f ( x ) = x − 1 x + 1 for x = 1 .
Find the y-intercept by setting x = 0 : f ( 0 ) = − 1 .
The correct answer is C: The y -intercept is y = − 1 . C
Explanation
Factor the function We are given the function f ( x ) = x 2 − 2 x + 1 x 2 − 1 and asked to determine which statement about its graph is correct. The options involve holes, vertical asymptotes, the y-intercept, and horizontal asymptotes. To analyze the function, we first factor the numerator and denominator.
Simplify the function Factoring the numerator and denominator, we have f ( x ) = ( x − 1 ) ( x − 1 ) ( x − 1 ) ( x + 1 ) = ( x − 1 ) 2 ( x − 1 ) ( x + 1 ) We can simplify this expression by canceling the common factor of ( x − 1 ) from the numerator and denominator, provided x = 1 :
f ( x ) = x − 1 x + 1 , x = 1
Find holes and vertical asymptotes Since we canceled a factor of ( x − 1 ) from both the numerator and denominator, there is a hole in the graph at x = 1 . The simplified function is f ( x ) = x − 1 x + 1 .
To find the vertical asymptote, we set the denominator of the simplified function equal to zero: x − 1 = 0 ⇒ x = 1 However, we already know there is a hole at x = 1 , so there is no vertical asymptote at x = 1 .
Find the y-intercept To find the y-intercept, we set x = 0 in the simplified function: f ( 0 ) = 0 − 1 0 + 1 = − 1 1 = − 1 So the y-intercept is y = − 1 .
Find the horizontal asymptote To find the horizontal asymptote, we consider the limit of the function as x approaches infinity: x → ∞ lim x − 1 x + 1 = x → ∞ lim 1 − x 1 1 + x 1 = 1 − 0 1 + 0 = 1 So there is a horizontal asymptote at y = 1 .
Check the statements Now we check the given statements: A. There is a hole at x = − 1 . This is incorrect, as there is a hole at x = 1 .
B. There is a vertical asymptote at x = − 1 . This is incorrect, as there is a vertical asymptote at x = 1 before considering the hole. C. The y -intercept is y = − 1 . This is correct, as we found f ( 0 ) = − 1 .
D. There is a horizontal asymptote at y = − 1 . This is incorrect, as there is a horizontal asymptote at y = 1 .
Therefore, the correct statement is C.
Final Answer The correct statement is that the y -intercept is y = − 1 .
Examples
Understanding the behavior of rational functions, like the one in this problem, is crucial in many real-world applications. For example, in physics, the function might describe the relationship between distance and time for an object with variable acceleration. Identifying asymptotes and intercepts helps predict the object's long-term behavior and initial conditions. Similarly, in economics, such functions can model cost-benefit ratios, where asymptotes indicate saturation points or limits of growth, and intercepts show initial investments or returns. By analyzing these functions, we can make informed decisions and predictions in various fields.
The correct answer is C: The y-intercept is y = − 1 . The analysis shows that the function has a hole at x=1, a vertical asymptote (which is actually not present due to the hole), and a horizontal asymptote at y=1. The y-intercept, found by evaluating f(0), is indeed -1.
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