Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.

Step-by-Step Solution

Answer

The product of an odd function and an even function is odd, as proved from the given exercise.

Step 1: Exemplify with Functions

Start solving the problem by taking examples of odd and even functions. Let's consider \(f(x) = x\) as an even function and \(g(x) = x^2\) as an odd function. Multiply \(f(x) \times g(x) = x \times x^2 = x^3\). Here, \(x^3\) is an odd function.

Step 2: Form a Hypothesis

Based on the example, we can hypothesize that the product of an even and odd function gives an odd function.

Step 3: Prove the Hypothesis

To prove this hypothesis, consider an even function \(f(x)\) and an odd function \(g(x)\). In general, an even function satisfies \(f(x) = f(-x)\), and an odd function satisfies \(g(x) = -g(-x)\). So, when multiplied together, \(f(x) \times g(x) = f(-x) \times -g(-x) = -f(-x) \times g(-x)\), which means the resulting function is odd.