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.

Key concepts

Function Decomposition

Function decomposition is a vital concept in mathematics where a complex function is expressed as a combination of simpler functions. Understanding function decomposition helps in dividing intricacies of a function into more manageable parts. This not only aids in the analysis of the function but also in performing simpler calculations. In the context of the exercise, function decomposition revolves around expressing any function \( f(x) \) as the sum of an even function \( g(x) \) and an odd function \( h(x) \).

• \( f(x) = g(x) + h(x) \) is the starting assumption.
• Our aim is to identify appropriate expressions for \( g(x) \) and \( h(x) \) based on \( f(x) \).

By understanding how functions can be broken down into these even and odd parts, it becomes easier to work with functions in different mathematical applications.

Proof by Example

Proof by example is a method that demonstrates the validity of a statement by using a specific example. In our exercise, we assume a general function \( f(x) \) and show how it can be expressed as a sum of an even and an odd function. Here, steps to prove involve algebraic manipulation to derive \( g(x) \) and \( h(x) \).- Start with known properties of even and odd functions: \( g(x) = g(-x) \) and \( h(x) = -h(-x) \).- Use these properties to form equations with \( f(x) \) and \( f(-x) \).- Through the example provided, we derived the expressions for \( g(x) \) and \( h(x) \): \[ g(x) = \frac{1}{2}(f(x) + f(-x)) \] \[ h(x) = \frac{1}{2}(f(x) - f(-x)) \]- Verify these to ensure \( g(x) \) is even and \( h(x) \) is odd.

Even Function Properties

Even functions are unique due to their symmetric nature about the y-axis. An even function \( g(x) \) fulfills the condition \( g(x) = g(-x) \) for all x in its domain.- **Symmetric Graph**: The graph of \( g(x) \) mirrors itself across the y-axis.- **Expression**: The even part of a function \( f(x) \) can be expressed as: \[ g(x) = \frac{1}{2}(f(x) + f(-x)) \]- **Role in Decomposition**: In function decomposition, \( g(x) \) captures all the symmetrical aspects of \( f(x) \) relative to the y-axis.These properties help in recognizing even functions and utilizing them in solving problems related to symmetry and function decomposition.

Odd Function Properties

Odd functions are characterized by their point symmetry about the origin. An odd function \( h(x) \) satisfies the condition \( h(x) = -h(-x) \) for all x within its domain.- **Point Symmetry**: The graph of an odd function is rotationally symmetric about the origin.- **Expression**: The odd part of a function \( f(x) \) can be represented as: \[ h(x) = \frac{1}{2}(f(x) - f(-x)) \]- **Role in Decomposition**: The odd function \( h(x) \) contains aspects of \( f(x) \) that are anti-symmetric about the origin.Understanding these properties aids in identifying odd functions and effectively using them to dissect general functions into simpler forms.