Give algebraic proofs that for even and odd functions: (a) even times even \(=\) even; odd times odd \(=\) even; even times odd \(=\) odd; (b) the derivative of an even function is odd; the derivative of an odd function is even.

In mathematics, functions can be classified based on their symmetry properties. An **even function** has the characteristic that for every input value x, the function value at x is the same as the function value at -x, symbolically written as: \(f(x) = f(-x)\). This means that the graph of an even function is symmetric with respect to the y-axis.

An **odd function**, on the other hand, has the property that for every input value x, the function value at x is the negative of the function value at -x, described as: \(f(x) = -f(-x)\). This means that the graph of an odd function is symmetric with respect to the origin. Understanding these properties is essential as they help identify and analyze the behavior of different functions.

Proving the properties of even and odd functions often involves algebraic manipulation. Let's explore some key proofs.

**Even Times Even:** Suppose \(f(x)\) and \(g(x)\) are even functions. Then \(f(x) = f(-x)\) and \(g(x) = g(-x)\). Consider the product function \(h(x) = f(x)g(x)\). To find out if \(h(x)\) is even, evaluate \(h(-x)\):

\(h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)\). Hence, \(h(x)\) is even.

**Odd Times Odd:** Suppose \(f(x)\) and \(g(x)\) are odd functions. Then \(f(x) = -f(-x)\) and \(g(x) = -g(-x)\). Consider the product function \(h(x) = f(x)g(x)\). To find out if \(h(x)\) is even, evaluate \(h(-x)\):

\(h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x)\). Therefore, \(h(x)\) is even.

**Even Times Odd:** Suppose \(f(x)\) is even and \(g(x)\) is odd. Then \(f(x) = f(-x)\) and \(g(x) = -g(-x)\). Consider the product function \(h(x) = f(x)g(x)\). To find out if \(h(x)\) is odd, evaluate \(h(-x)\):

\(h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -h(x)\). Hence, \(h(x)\) is odd.

When working with multiple functions, understanding how their properties combine is crucial. Let's explore the results of multiplying even and odd functions.

**Even Function Times Even Function:**
If you have two even functions, their product will also be an even function. This is because the property of being symmetric about the y-axis is retained.

**Odd Function Times Odd Function:**
The product of two odd functions is always an even function. This is due to the symmetry about the origin that cancels out the negative signs from each function.

**Even Function Times Odd Function:**
If you multiply an even function by an odd function, the resulting product is an odd function. The combination retains the asymmetry about the origin due to the influence of the odd function.

Whenever you encounter functions where their product properties are tested, use their definitions to understand why these rules apply.

Taking the derivative of an even or odd function results in specific outcomes due to their intrinsic properties.

**Derivative of an Even Function:**
Let \(f(x)\) be an even function, so \(f(x) = f(-x)\). The derivative, denoted by \(f'(x)\), can be analyzed by evaluating \(f'(-x)\):

Using the chain rule, \(f'(-x) = -f'(x)\), indicating that \(f'(x)\) is an odd function.

**Derivative of an Odd Function:**
Let \(f(x)\) be an odd function, so \(f(x) = -f(-x)\). The derivative, denoted by \(f'(x)\), can be analyzed by evaluating \(f'(-x)\):

Using the chain rule, \(f'(-x) = f'(x)\), indicating that \(f'(x)\) is an even function.

These specific properties of derivatives are handy when you need to understand the behavior or characteristics of a function after differentiation. Always use the definitions and symmetry properties to validate these outcomes.