Final answer:
The function f(x) = (3x-1)/(x+2) has a vertical asymptote at x = -2 and a horizontal asymptote at y = 3. The vertical asymptote is found where the denominator is zero, and the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
Step-by-step explanation:
To find the vertical and horizontal asymptotes of the function f(x) = (3x-1)/(x+2), we need to examine the function's behavior as the variable x approaches certain values.
For the vertical asymptote, we look for values of x that would make the denominator of our function equal to zero, as the function tends toward infinity.
In this case, setting x+2 equal to zero gives us x = -2, which is the equation of a vertical line and thus a vertical asymptote.
There is no other value of x that would make the denominator zero, so x = -2 is the only vertical asymptote.
To find a horizontal asymptote, we compare the degrees of the numerator and the denominator.
Since both the numerator and the denominator are of degree 1 (the highest power of x is x to the first power), the horizontal asymptote is found by the ratio of the leading coefficients.
Here that would be 3/1, so y = 3 is the horizontal asymptote.
As x goes to positive or negative infinity, the function f(x) approaches y = 3.
Therefore, the function f(x) = (3x-1)/(x+2) has a vertical asymptote at x = -2 and a horizontal asymptote at y = 3.
Complete Question:
Find all vertical and horizontal asymptotes of the graph of the given function.
f(x) = (3x-1)/(x+2)