Question

Prove that any function can be expressed as the sum of two other functions, one of which is even and the other odd. That is, for any function \(f,\) whose domain contains \(-x\) whenever it contains \(x,\) show that there is an even function \(g\) and an odd function \(h\) such that \(f(x)=g(x)+h(x)\) Hint: Assuming \(f(x)=g(x)+h(x),\) what is \(f(-x) ?\)

Short answer

#Question#: Prove that for any function f, we can express it as the sum of two functions g and h, where g is an even function and h is an odd function.

Step-by-step solution

Definitions of even and odd functions

Even functions have the property that \(g(x) = g(-x)\) for all x in the domain. Odd functions have the property that \(h(x) = -h(-x)\) for all x in the domain. We are given the hint to consider the expression for \(f(-x)\), assuming \(f(x) = g(x) + h(x)\).

Express g(x) and h(x) in terms of f(x) and f(-x)

First, note that the function f has the property that its domain contains both x and -x. Assuming \(f(x) = g(x) + h(x)\), let's find \(f(-x)\): $$f(-x) = g(-x) + h(-x)$$ Now, let's add the expressions of \(f(x)\) and \(f(-x)\): $$f(x) + f(-x) = (g(x) + h(x)) + (g(-x) + h(-x))$$ Then, let's subtract the expressions of \(f(x)\) and \(f(-x)\): $$f(x) - f(-x) = (g(x) + h(x)) - (g(-x) + h(-x))$$ Notice that the terms in these equations can be regrouped to isolate g(x) and h(x): $$g(x) + g(-x) = f(x) + f(-x)$$ $$h(x) - h(-x) = f(x) - f(-x)$$ Now we can solve for g(x) and h(x) in terms of f(x) and f(-x): $$g(x) = \frac{1}{2}(f(x) + f(-x))$$ $$h(x) = \frac{1}{2}(f(x) - f(-x))$$

Demonstrating g and h as even and odd functions

Next, let's prove g(x) is even by showing that \(g(x) = g(-x)\): $$g(-x) = \frac{1}{2}(f(-x) + f(x)) = g(x)$$ Similarly, let's prove h(x) is odd by showing that \(h(x) = -h(-x)\): $$h(-x) = \frac{1}{2}(f(-x) - f(x)) = -h(x)$$ This confirms that g is an even function and h is an odd function.

Expressing f(x) as the sum of g(x) and h(x)

Finally, since we have derived g(x) and h(x) from f(x) and f(-x), we can rewrite f(x) as the sum of g(x) and h(x): $$f(x) = g(x) + h(x) = \frac{1}{2}(f(x) + f(-x)) + \frac{1}{2}(f(x) - f(-x))$$ Therefore, we have successfully proved that any function f can be expressed as the sum of two other functions g and h, where g is an even function and h is an odd function.