How do you graph #f(x)=(x-1)/(x+1)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

Vertical asymptote: #x = -1#
Horizontal asymptote: #y = 1#
No holes, #x#-intercept #(1, 0)#; #y#-intercept #(0, -1)#

Rational equation: #f(x)=(N(x))/(D(x)) = (a_nx^n+...)/(b_mx^m+...)#
Find x-intercepts #N(x) = 0#: #x-1 = 0; x = 1# #x#-intercept #(1, 0)#
Find y-intercepts Set #x = 0#: #f(0) = -1/1 = -1# #y#-intercept #(0, -1)#

Find holes: Holes occur when factors can be cancelled because they are found both in the numerator and denominator. This does not occur in this problem.

Find the vertical asymptotes #D(x) = 0#: Vertical asymptotes at #x +1 = 0; x = -1#
Find horizontal asymptotes When #m=n, y = a_n/b_m#: #m = n= 1# so there is a horizontal asymptote at #y = 1#

graph{(x-1)/(x+1) [-10, 10, -5, 5]}

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Answer 2

To graph the function f(x) = (x-1)/(x+1), we can start by identifying the vertical and horizontal asymptotes, holes, x-intercepts, and y-intercept.

Vertical asymptote: The vertical asymptote occurs when the denominator of the function is equal to zero. In this case, x+1 = 0, so x = -1 is the vertical asymptote.

Horizontal asymptote: To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator. In this case, both the numerator and denominator have a degree of 1. Since the degrees are the same, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). Therefore, the horizontal asymptote is y = 1/1, which simplifies to y = 1.

Hole: To find the hole, we need to factor the numerator and denominator and cancel out any common factors. In this case, the numerator x-1 and the denominator x+1 do not have any common factors. Therefore, there is no hole in this function.

X-intercept: To find the x-intercept, we set y = 0 and solve for x. In this case, (x-1)/(x+1) = 0. Since the numerator can only be zero when x = 1, the x-intercept is x = 1.

Y-intercept: To find the y-intercept, we set x = 0 and solve for y. In this case, f(0) = (0-1)/(0+1) = -1. Therefore, the y-intercept is y = -1.

Now, we can plot these points and draw the graph accordingly.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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