If f(x) is a polynomial function, does 1/f(x) always have a horizontal asymptote? Explain why or why not and provide a counterexample if needed.
To determine whether 1/f(x) always has a horizontal asymptote, we need to consider the behavior of the polynomial function f(x) and evaluate its degree.
If the degree of f(x) is greater than zero, then f(x) will have at least one term with a nonzero coefficient in the highest degree. In this case, as x approaches positive or negative infinity, f(x) will also tend towards positive or negative infinity respectively. Consequently, 1/f(x) will tend towards zero as x approaches infinity or negative infinity, implying that it does have a horizontal asymptote at y = 0.
However, if the degree of f(x) is zero (i.e., f(x) is a constant), then f(x) has no horizontal asymptote. In this scenario, since f(x) is a constant, 1/f(x) will also be a constant. Hence, it does not approach any particular value as x approaches infinity or negative infinity, and therefore 1/f(x) does not have a horizontal asymptote.
To provide a counterexample, let's consider the polynomial function f(x) = 1. In this case, f(x) is a constant function, and its reciprocal 1/f(x) is also constant. Since it does not approach any value as x approaches infinity or negative infinity, 1/f(x) does not have a horizontal asymptote.
In conclusion, 1/f(x) always has a horizontal asymptote at y = 0 if f(x) is a polynomial function of degree greater than zero. However, if f(x) is a constant function, 1/f(x) does not have a horizontal asymptote.