Determine the vertical asymptotes of the graph of the following functions
f(x)=5+2x^2/2-x-x^2
f(x)=x^2-25/x^3-6x^2+5x
To find the vertical asymptotes of a rational function, we need to determine the values of x that make the denominator equal to zero.
For the first function, f(x) = (5 + 2x^2)/(2 - x - x^2):
Setting the denominator equal to zero, we have:
2 - x - x^2 = 0
Rearranging the terms, we get:
x^2 + x - 2 = 0
Factoring, we have:
(x + 2)(x - 1) = 0
Setting each factor equal to zero, we get:
x + 2 = 0 or x - 1 = 0
x = -2 or x = 1
So the vertical asymptotes of the first function are x = -2 and x = 1.
For the second function, f(x) = (x^2 - 25)/(x^3 - 6x^2 + 5x):
Setting the denominator equal to zero, we have:
x^3 - 6x^2 + 5x = 0
Factoring out an x, we get:
x(x^2 - 6x + 5) = 0
The quadratic equation x^2 - 6x + 5 = 0 does not factor easily. However, we can solve it using the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(1)(5)))/(2(1))
x = (6 ± √(36 - 20))/2
x = (6 ± √16)/2
x = (6 ± 4)/2
x = 5 or x = 1
So the vertical asymptotes of the second function are x = 0, x = 5, and x = 1.
To determine the vertical asymptotes of a function, we need to find the values of x for which the function approaches infinity or negative infinity.
1. For f(x) = (5 + 2x^2) / (2 - x - x^2):
First, we need to find the values of x that make the denominator equal to zero since a vertical asymptote occurs when the denominator is equal to zero.
Set the denominator equal to zero and solve for x:
2 - x - x^2 = 0
This is a quadratic equation. Rearranging and factoring, we have:
-x^2 - x + 2 = 0
(-x + 2)(x + 1) = 0
Setting each factor equal to zero gives us two possible values for x:
-x + 2 = 0 --> x = 2
x + 1 = 0 --> x = -1
Thus, the vertical asymptotes for f(x) are x = 2 and x = -1.
2. For f(x) = (x^2 - 25) / (x^3 - 6x^2 + 5x):
We need to find the values of x that make the denominator equal to zero.
Set the denominator equal to zero and solve for x:
x^3 - 6x^2 + 5x = 0
This is also a quadratic equation. We can factor out an x to simplify:
x(x^2 - 6x + 5) = 0
Setting each factor equal to zero gives us three possible values for x:
x = 0
x^2 - 6x + 5 = 0
To solve the quadratic equation, we can factor:
(x - 5)(x - 1) = 0
Setting each factor equal to zero gives us two more possible values for x:
x - 5 = 0 --> x = 5
x - 1 = 0 --> x = 1
Thus, the vertical asymptotes for f(x) are x = 0, x = 1, and x = 5.