To find the vertical and horizontal asymptotes of the function f(x) = x² + 6x + 3, we need to analyze the function’s behavior as x approaches certain limits.
Vertical Asymptotes
Vertical asymptotes occur where the function tends to infinity or negative infinity, usually where the denominator is zero in a rational function. However, in this case, f(x) is a polynomial function and does not have a denominator. Therefore, there are no vertical asymptotes for this function.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. For polynomial functions, we compare the degrees of the numerator and the denominator. Since there is no denominator here, we look at the degree of the polynomial:
- The degree of the numerator (the highest power of x in f(x)) is 2.
- Since there is no denominator, we consider the leading term of the polynomial as x approaches infinity or negative infinity.
As x → ±∞, f(x) = x² will dominate. Therefore, the function tends to infinity. This means:
There are no horizontal asymptotes for this function since it does not approach any finite value as x tends to infinity or negative infinity.
In summary, the function f(x) = x² + 6x + 3 has no vertical asymptotes and no horizontal asymptotes.