given the following rational function
a. state the domain
b. find the vertical and horizontal asymptotes if any
c. find the oblique asymptote
f(x) = x^2 + 7x + 5 / x + 8
as usual, the domain is all reals except where the denominator is zero. That is, all reals except x = -8
Vertical asymptotes occur when the denominator is zero and the numerator is not.
at x = -8, the numerator is not zero, so that's the vertical asymptote.
For large x, f(x) is just x^2/x = x, so there's an oblique asymptote at the line y=x, and here are no horizontal asymptotes.
To determine the domain, vertical and horizontal asymptotes, and the oblique asymptote of the given rational function f(x) = (x^2 + 7x + 5) / (x + 8), follow these steps:
a. Domain:
The domain of a rational function is the set of all real numbers except the values that make the denominator zero. In this case, the denominator is x + 8, so we need to find the values that make it zero:
x + 8 = 0
x = -8
Therefore, the domain of f(x) is all real numbers except -8. In interval notation, it can be written as (-∞, -8) ∪ (-8, +∞).
b. Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function is zero but the numerator is not zero at that point. In this case, the denominator is x + 8, so we need to find the value(s) of x that make(s) the denominator zero:
x + 8 = 0
x = -8
Therefore, there is a vertical asymptote at x = -8.
c. Horizontal Asymptotes:
Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote.
d. Oblique Asymptote:
To find the oblique asymptote, divide the numerator by the denominator using long division or synthetic division and see what the result approaches as x approaches positive or negative infinity.
Using long division, divide x^2 + 7x + 5 by x + 8:
x + 6 (quotient)
x + 8 | x^2 + 7x + 5
- (x^2 + 8x)
- x + 5
- ( - x - 8)
13 (remainder)
The quotient is x + 6, and the remainder is 13. Therefore, the oblique asymptote is y = x + 6.
In summary:
a. The domain is all real numbers except x = -8: (-∞, -8) ∪ (-8, +∞).
b. There is a vertical asymptote at x = -8.
c. There is no horizontal asymptote.
d. The oblique asymptote is y = x + 6.