Solve the asymptotes
x
x2 − 4
3x
x2 − x − 6
1
x
5
x2
To solve for the asymptotes of a function, you need to consider two types of asymptotes: vertical asymptotes and horizontal asymptotes.
1. Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function equals zero. To find vertical asymptotes, set the denominator equal to zero and solve for the values of x.
For the first equation, x / (x^2 - 4), the denominator is x^2 - 4. Setting x^2 - 4 equal to zero, you have:
x^2 - 4 = 0
This is a quadratic equation that can be factored as (x - 2)(x + 2) = 0.
Solving this equation, you get x = 2 and x = -2.
Therefore, the vertical asymptotes for the first equation are x = 2 and x = -2.
For the second equation, 3x / (x^2 - x - 6), the denominator is x^2 - x - 6. Setting x^2 - x - 6 equal to zero, you have:
x^2 - x - 6 = 0
This can be factored as (x - 3)(x + 2) = 0.
Solving this equation, you get x = 3 and x = -2.
Therefore, the vertical asymptotes for the second equation are x = 3 and x = -2.
For the third equation, 1 / x, the denominator is simply x. There are no values of x that make the denominator zero.
Therefore, there are no vertical asymptotes for the third equation.
2. Horizontal Asymptotes:
Finding horizontal asymptotes requires analyzing the degrees of the numerator and denominator of the function.
For the first equation, x / (x^2 - 4), the degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0.
For the second equation, 3x / (x^2 - x - 6), the degree of the numerator is 1 and the degree of the denominator is also 1. Since the degrees are the same, you need to compare the coefficients of the highest degree terms. The coefficient of x^2 is 1 in both the numerator and denominator. Dividing the coefficients, you have 3/1 = 3.
Therefore, the horizontal asymptote for the second equation is y = 3.
For the third equation, 5 / x^2, the degree of the numerator is 0 (since it is a constant) and the degree of the denominator is 2. Since the degree of the denominator is greater, the horizontal asymptote is at y = 0.
In conclusion:
- First equation: Vertical asymptotes at x = 2 and x = -2, and horizontal asymptote at y = 0.
- Second equation: Vertical asymptotes at x = 3 and x = -2, and horizontal asymptote at y = 3.
- Third equation: No vertical asymptotes, and horizontal asymptote at y = 0.