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Chapter 2: Problem 118

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.

Short Answer

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

Step by step solution

01

- 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.
02

- 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, \(fg(-x) = f(-x)g(-x)\).
03

- 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.
04

- 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.

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