How do you graph #f(x)=-3/x# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Eva Abramson · Accepted Answer

See graph

no holes.

Set #x=0# to solve for #y# intercept: #f(x)=-3/x# #-3/0# in undefined so no #y# intercept exists Set #f(x)=y=0# to solve for #x# intercept(s): #0=-3/x# #0=-3# is not possible so no #x# intercept(s) exist. There are no holes because you cannot cancel any factors with #x# in them from the denominator. Set the denominator=0 to solve fo asymptotes: #x=0# so there is a vertical asymptote at #y=0#. #x -> +-oo, f(x) -> 0# so there is a horizontal asymptote at #x=0# graph{-3/x [-10, 10, -5, 5]}

Eli Assouline · Answer

undefined To graph the function f(x) = -3/x, we can follow these steps:

1. Holes: The function has no holes.

2. Vertical Asymptotes: Set the denominator equal to zero and solve for x. In this case, x cannot be zero since division by zero is undefined. Therefore, there is a vertical asymptote at x = 0.

3. Horizontal Asymptotes: Determine the behavior of the function as x approaches positive or negative infinity. As x approaches positive infinity, the function approaches 0. As x approaches negative infinity, the function also approaches 0. Hence, there is a horizontal asymptote at y = 0.

4. x-intercept: To find the x-intercept, set f(x) equal to zero and solve for x. In this case, -3/x = 0. Since the numerator is never zero, there are no x-intercepts.

5. y-intercept: To find the y-intercept, set x equal to zero and evaluate f(x). In this case, f(0) = -3/0, which is undefined. Therefore, there is no y-intercept.

By considering these aspects, you can accurately graph the function f(x) = -3/x.