To find the vertical and horizontal asymptotes of the function f(x) = 3x2 – 2x + 4, we first need to analyze the function itself.
Vertical Asymptotes
Vertical asymptotes occur when the function approaches infinity or negative infinity as the input approaches a certain value. For rational functions, these often occur at values that make the denominator zero. However, since our function is a polynomial, which is smooth and continuous everywhere, there are no vertical asymptotes.
Horizontal Asymptotes
To find horizontal asymptotes, we examine the behavior of the function as x approaches infinity or negative infinity. For the function f(x) = 3x2 – 2x + 4, the leading term (which determines the horizontal asymptote) is 3x2.
As x approaches positive or negative infinity, the term 3x2 dominates, and the function approaches infinity. Therefore, there are no horizontal asymptotes since the function does not level off at any finite value as x goes to infinity.
In summary, for the function f(x) = 3x2 – 2x + 4:
- No vertical asymptotes
- No horizontal asymptotes
Thus, the function is continuous without any asymptotic behavior in either the vertical or horizontal direction.