A continuous function, f, has domain all real numbers. If f(-1) = 5 and f(1) = -5, explain why f must have at least one zero in the interval [-1, 1].

To understand why the function f must have at least one zero in the interval [-1, 1], let's use the Intermediate Value Theorem.

The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b], and if k is any value between f(a) and f(b), then there exists at least one number c in the interval [a, b] such that .

In this case, we have a continuous function f with domain all real numbers, and f(-1) = 5 and f(1) = -5. Since the interval [-1, 1] is a closed interval, we can apply the Intermediate Value Theorem to this interval.

Let's set k = 0. According to the Intermediate Value Theorem, there must exist at least one number c in the interval [-1, 1] such that f(c) = 0, since 0 is between f(-1) and f(1).

Therefore, f must have at least one zero in the interval [-1, 1].