Question

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)`

Explanation:

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]}