To determine whether the product of two even functions, f and g, is also an even function, let’s start by recalling the definition of an even function.
A function h(x) is considered even if it satisfies the following condition for all x in its domain:
h(-x) = h(x)
Now, since both f and g are given as even functions, we can express this property as:
f(-x) = f(x) ext{ and } g(-x) = g(x)
Next, we need to analyze the product function fg(x):
(fg)(-x) = f(-x)g(-x)
Using the property of even functions we noted earlier, this becomes:
(fg)(-x) = f(x)g(x) = (fg)(x)
From this derivation, we can conclude that the product of the two even functions f and g is indeed an even function, since it satisfies the condition:
fg(-x) = fg(x)
In summary, if both f and g are even functions, their product fg is also an even function.