Enter the equations of the asymptotes for the function f(x) .
f(x)= −(2/x+4) − 6
Vertical Asymptote: ?
Horizontal Asymptote: ?
when x = -4, there is a zero in the denominator (vert)
as x goes to ±∞, the 1st term of the expression goes to zero, leaving -6 (horiz)
Vertical asymptote: -4
Horizontal asymptote: -6
To find the equations of the asymptotes for the function f(x), you need to consider the behavior of the function as x approaches infinity or negative infinity.
1. Vertical Asymptote:
The vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator is x + 4. So, set x + 4 = 0 and solve for x.
x + 4 = 0
x = -4
Therefore, the vertical asymptote for the function f(x) is x = -4. The equation of the vertical asymptote is x = -4.
2. Horizontal Asymptote:
To find the horizontal asymptote, you need to compare the degrees of the highest power terms in the numerator and the denominator.
- Since the degree of the highest power term in the numerator is 0 (constant), the numerator does not grow as x approaches infinity or negative infinity.
- The degree of the highest power term in the denominator is 1, which is x.
- When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0.
Therefore, the horizontal asymptote for the function f(x) is y = 0. The equation of the horizontal asymptote is y = 0.
In summary:
Vertical Asymptote: x = -4
Horizontal Asymptote: y = 0