What are the x intercept(s), y intercept, symetry, verticle asymtote(s), and horizontal or oblique asymtotes?

Question

5x all over x^2-1

Answer 1

Remember that all functions are in the form f(x) = [something] or y = [something, so I presume you meant the bottom function (or something like it):

y = 5x/(x^2-1)
or
f(x) = 5x/(x^2-1)

To find the x-intercepts, you have to make y equal to 0.

0 = 5x/(x^2-1)
0 = 5x
x = 0

Therefore, 0 is an x-intercept.

To find the y-intercept(s), you make x = 0.

y = 5(0)/(0^2 - 1) = 0

Therefore, 0 is a y-intercept.

To calculate symmetry, we have to test if the function is even or odd. If f(x) = f(-x), then the function is even. But
f(-x) = -5x/(x^2 - 1), which is not equal to f(x), so it's not even.
To test if it is odd, we have to determine whether -f(x) = f(-x)
-f(x) = -5x/(x^2 - 1) = f(-x)
So the function has odd symmetry.

The horizontal asymptotes are found by taking the limit to infinity.

lim (x -> infinity, 5x/(x^2 - 1)) = [divide each term by the highest power, x^2]

lim (x -> infinity, (5/x) / (1 - 1/(x^2))) = 0/(1 - 0) = 0

So y = 0 is a horizontal asymptote.

Answer 2

y = 5x / (x²-1)

a)
x intercept
y = 5(0) / (0²-1)
y = 0

Px = (0, 0)

b)
y intercept
0 = 5x / (x²-1)
5x = 0
x = 0

Py = (0, 0)

c)
verticle asymtote(s)
x = -1 e x = 1

d)
horizontal asymtote
y = 0

₢

Answer 3

there are also 2 vert asymptotes at x=-1 & x=+1