Which of the following rational functions is graphed below?
The graph below shows a rational function. Which of the following rational functions is represented by this graph?
y = f(x) To answer this question, we can use the following steps:
- Identify the vertical asymptotes. The vertical asymptotes of a rational function are the values of x where the denominator of the function is equal to zero. In this case, the denominator of the function is (x-1)(x+1). Therefore, the vertical asymptotes are x=1 and x=-1.
- Identify the horizontal asymptote. The horizontal asymptote of a rational function is the value of y as x approaches infinity. In this case, the degree of the numerator is equal to the degree of the denominator. Therefore, the horizontal asymptote is y=0.
- Identify the behavior of the function near the vertical asymptotes. The function will approach the vertical asymptotes as x approaches the vertical asymptotes from either side. In this case, the function will approach infinity as x approaches 1 from the left and right. The function will also approach infinity as x approaches -1 from the left and right.
- Identify the behavior of the function away from the vertical asymptotes. The function will approach the horizontal asymptote as x approaches infinity. In this case, the function will approach 0 as x approaches infinity.
Based on these steps, we can conclude that the following rational function is represented by the graph:
y = f(x) = 1 / (x-1)(x+1) This function has vertical asymptotes at x=1 and x=-1. It has a horizontal asymptote at y=0. The function approaches infinity as x approaches 1 or -1. The function approaches 0 as x approaches infinity.
Additional questions
- What is the domain of the function?
The domain of a rational function is the set of all real numbers x for which the denominator of the function is not equal to zero. In this case, the denominator of the function is (x-1)(x+1). Therefore, the domain of the function is all real numbers x except for 1 and -1.
- What is the range of the function?
The range of a rational function is the set of all possible values of y for the function. In this case, the function approaches the horizontal asymptote y=0 as x approaches infinity. Therefore, the range of the function is all real numbers less than or equal to 0.
- What are the critical points of the function?
The critical points of a function are the values of x where the derivative of the function is equal to zero or undefined. In this case, the derivative of the function is
f'(x) = (1)(x+1) - (1)(x-1) / (x-1)(x+1)^2 This derivative is equal to zero when x=-1. The derivative is undefined when x=1. Therefore, the only critical point of the function is x=-1.
- What are the increasing/decreasing intervals of the function?
The increasing/decreasing intervals of a function are the intervals where the function is increasing or decreasing. To find these intervals, we can use the sign of the derivative of the function.
The derivative of the function is positive for x<-1 and for x>1. The derivative of the function is negative for -1<x<1. Therefore, the function is increasing for x<-1 and for x>1. The function is decreasing for -1<x<1.
- What are the relative extrema of the function?
The relative extrema of a function are the points on the graph of the function where the direction of the function changes from increasing to decreasing or vice versa. In this case, the function changes from increasing to decreasing at x=-1. Therefore, the function has a relative maximum at x=-1.
I hope this helps!
