Show that sum of two even function with the same domain is an even function.
f(x) = f(-x)
g(x) = g(-x)
h(x) = f(x) + g(x)
h(-x) = f(-x)+g(-x) = f(x)+g(x)
so
h(-x)=h(x)
Thanks!!
To prove that the sum of two even functions is also an even function, we need to show that it satisfies the definition of an even function.
Let's say we have two even functions, f(x) and g(x), with the same domain.
By definition, an even function satisfies the property f(x) = f(-x) for all x in its domain.
To prove that the sum of two even functions, h(x) = f(x) + g(x), is even, we need to show that h(x) = h(-x) for all x in the domain.
Now let's consider h(x) = f(x) + g(x).
To prove h(x) is even, we need to show that h(x) = h(-x) for any value of x in the domain.
Let's substitute -x into h(x):
h(-x) = f(-x) + g(-x)
Since f(x) is an even function, we know that f(-x) = f(x):
h(-x) = f(x) + g(-x)
Similarly, since g(x) is an even function, we know that g(-x) = g(x):
h(-x) = f(x) + g(x)
And this is equal to h(x) by definition.
Therefore, we have shown that the sum of two even functions with the same domain is an even function, as h(x) = h(-x) for all x in the domain.