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].
since f is continuous, there are no holes in the graph. Since f(-1) > 0 and f(1) < 0, f must assume all values between 5 and -5. That includes zero.
To explain why f must have at least one zero in the interval [-1, 1], we can use the Intermediate Value Theorem.
The Intermediate Value Theorem states that if a continuous function f(x) is defined on a closed interval [a, b], and f(a) and f(b) have opposite signs (one positive and the other negative), then there exists at least one value c in the interval (a, b) such that f(c) = 0.
In the given problem, we have a continuous function f with domain of all real numbers. We are also given that f(-1) = 5 and f(1) = -5. Since f(-1) is positive and f(1) is negative, we have f(-1) > 0 and f(1) < 0.
Because f is continuous, it means that it does not have any jumps, breaks, or holes. So, it must pass through all values in between f(-1) = 5 and f(1) = -5. By the Intermediate Value Theorem, there must exist at least one value c in the interval (-1, 1) such that f(c) = 0, since f(-1) > 0 and f(1) < 0.
Therefore, we can conclude that f must have at least one zero in the interval [-1, 1].
To explain why the function f must have at least one zero in the interval [-1, 1], we can make use of the Intermediate Value Theorem.
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and f(a) and f(b) are of different signs (meaning f(a) is positive and f(b) is negative, or vice versa), then there exists at least one value c in the interval (a, b) where f(c) = 0.
Given the information that f(-1) = 5 and f(1) = -5, we can observe that these two function values have opposite signs. Since the function f is continuous over the entire real number domain, we can seamlessly apply the Intermediate Value Theorem to the interval [-1, 1].
Therefore, based on the Intermediate Value Theorem, there must exist at least one value c in the interval [-1, 1] where f(c) = 0.