Problem 118
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
Verified Answer
True, the product of two even functions is also an even function as established by the proof.
1Step 1: Define the functions
Let \(f\) and \(g\) be even functions, so it is true that \(f(-x) = f(x)\) and \(g(-x) = g(x)\) for all \(x\).
2Step 2: Consider the Product Function
We define \(h(x) = f(x)g(x)\). We need to prove that \(h(-x) = h(x)\) for all \(x\).
3Step 3: Prove the Product Function is Even
Evaluate \(h(-x)\) by substituting \(-x\) into the definition of \(h\): \[h(-x) = f(-x)g(-x).\] Since \(f\) and \(g\) are both even functions, this is equal to \[f(x)g(x) = h(x),\] which proves that \(h\) is even. Thus, the product of two even functions is also an even function.
Other exercises in this chapter
Problem 118
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