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

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.