suppose f and g are even functions what can you say about (f+g) and fg??please explain in details and thanks

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Matthew Fonda | eNotes Employee

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First, lets recall the definition of an even function. We say a function is even if

`f(x) = f(-x)`

for all x.

First, let's consider the case of the function h(x) = f(x) + g(x), where f and g are even functions. To show that h is also even, we need to show that h(x) = h(-x):

h(-x) = f(-x) + g(-x)

But f(-x) = f(x) and g(-x) = g(x)

=> h(-x) = f(x) + g(x)

=> h(-x) = h(x)

Therefore, h(x)  is even.

 

Next, we'll consider the case h(x) = f(x)g(x). Again, to prove this is even, we need to show that h(x) = h(-x)

h(-x)=f(-x)g(-x)

But f(-x) = f(x) and g(-x) = g(x)

=> h(-x) = f(x)g(x)

=> h(-x) = h(x)

Therefore h(x) is even.

If f and g are even functions, then the functions f+g and fg are also even.

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