Given: f(x) = (x+2)((x-1))/(x-3)
f(x) = (x+2)((x-1))/(x-3) = (x+2)/1((x-1))/(x-3) = ((x+2)(x-1))/(x-3)
This type of equation is called a rational (fraction) function.
It has the form: f(x) = (N(x))/(D(x)) = (a_nx^n + ...)/(b_m x^m + ...),
where N(x)) is the numerator and D(x) is the denominator,
n = the degree of N(x) and m = the degree of (D(x))
and a_n is the leading coefficient of the N(x) and
b_m is the leading coefficient of the D(x)
Step 1 factor : The given function is already factored.
Step 2, cancel any factors that are both in (N(x)) and D(x)) (determines holes):
The given function has no holes " "=> " no factors that cancel"
Step 3, find vertical asymptotes: D(x) = 0
vertical asymptote at x = 3
Step 4, find horizontal asymptotes:
Compare the degrees:
If n < m the horizontal asymptote is y = 0
If n = m the horizontal asymptote is y = a_n/b_m
If n > m there are no horizontal asymptotes
In the given equation: n = 2; m =1
There is no horizontal asymptotes
Step 5, find slant or oblique asymptote when n = m+1:
f(x) = ((x+2)(x-1))/(x-3) = (x^2 +x - 2)/(x-3)
Use Long division:
" "ul(" "color(red)(x + 4) +10/(x-3))
x-3 | x^2 + x - 2
" "ul(x^2 - 3x)
" "4x - 2
" "ul(4x - 12)
" "10
slant or oblique asymptote: color(red)(y = x+4)
Step 6, find x-intercept(s) : N(x) = 0
(x + 2)(x-1) = 0
" "x + 2 = 0 " "=> x"-intercept" (-2, 0)
" "x -1 = 0 " "=> x"-intercept" (1, 0)
Step 7, find y-intercept(s): x = 0
f(0) = (0+2)((0-1))/(0-3) = 2(-1)/-3 = 2/3
y-intercept: (0, 2/3)#
Graph of f(x) = (x+2)((x-1))/(x-3):
graph{(x+2)((x-1))/(x-3) [-50, 50, -50, 50]}
