The expected value is the sum of the x-values times the f(x) values.
An "even function" is defined as f(x) = f(-x). Here are some POSSIBLE and arbitrary values of x and f(x)
Notice that f(x) is the same whether x is a positive or negative value. That's the result of f(x) being an even function.
So, IF these were values of x and f(x), let's calculate the expected value:
EV = (1)(5) + (-1)(5) + (9)(8) + (-9)(8) + (27)(6) + (-27)(6) = 0
Although my choices for the values of x and f(x) are completely arbitrary, it is given that X is a continuous random variable. That means there will be as many values of x as there are -x and since EV = ∑x•f(x) we can expect that every x•f(x) will be "zeroed out" by an opposite (-x)•f(x).
I hope this helps.