Problem 118 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.

Step-by-Step Solution

Answer

True, the product of two even functions \(f\) and \(g\) is also an even function.

Step 1 - Define Even Function

In mathematics, an even function is a function that satisfies the property \(f(x) = f(-x)\) for every \(x\) in the function's domain.

Step 2 - Express \(fg\)

To show that \(fg\) is even, it is necessary to evaluate \(fg(-x)\). Since both \(f\) and \(g\) are even, \(f(-x) = f(x)\) and \(g(-x)=g(x)\). Thus, \(fg(-x) = f(-x)g(-x)\).

Step 3 - Apply the Definitions

Now using the fact that both \(f\) and \(g\) are even, we can rewrite the above equation: \(fg(-x) = f(x)g(x)\). This is the definition of an even function.

Step 4 - Conclusion

The expression derived in Step 3 proves the original statement: if \(f\) and \(g\) are even functions, their product \(fg\) will also be an even function.