Question

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.

Short answer

Even * even = even; Odd * odd = even; Even * odd = odd. Derivative of an even function is odd; Derivative of an odd function is even.

Step-by-step solution

- Define Even and Odd Functions

An even function satisfies the property: \[ f(-x) = f(x) \]An odd function satisfies the property: \[ g(-x) = -g(x) \]

- Prove Even Times Even is Even

Consider two even functions, \( f(x) \) and \( h(x) \). Then, \[ (f(x) \times h(x))(-x) = f(-x) \times h(-x) = f(x) \times h(x) \]This shows that the product of two even functions is even.

- Prove Odd Times Odd is Even

Consider two odd functions, \( g(x) \) and \( k(x) \). Then, \[ (g(x) \times k(x))(-x) = g(-x) \times k(-x) = (-g(x)) \times (-k(x)) = g(x) \times k(x) \]This shows that the product of two odd functions is even.

- Prove Even Times Odd is Odd

Consider an even function \( f(x) \) and an odd function \( g(x) \). Then, \[ (f(x) \times g(x))(-x) = f(-x) \times g(-x) = f(x) \times (-g(x)) = - (f(x) \times g(x)) \]This shows that the product of an even function and an odd function is odd.

- Derivative of Even Function is Odd

Let \( f(x) \) be an even function. Then, \( f(-x) = f(x) \). Differentiating both sides with respect to \( x \), we get \[ \frac{d}{dx}[f(-x)] = \frac{d}{dx}[f(x)] \]This simplifies to \[ -f'(-x) = f'(x) \]This shows that \( f'(x) \) is an odd function.

- Derivative of Odd Function is Even

Let \( g(x) \) be an odd function. Then, \( g(-x) = -g(x) \). Differentiating both sides with respect to \( x \), we get \[ \frac{d}{dx}[g(-x)] = \frac{d}{dx}[-g(x)] \]This simplifies to \[ -g'(-x) = -g'(x) \]Therefore, \[ g'(-x) = g'(x) \]This shows that \( g'(x) \) is an even function.

Key concepts

Even Functions

An even function is symmetrical around the y-axis. This means that if you reflect the graph of the function over the y-axis, it looks the same. Mathematically, an even function satisfies the equation:
\[ f(-x) = f(x) \]

This property tells us that the value of the function at any positive value of x is the same as the value at the corresponding negative value. For example, the functions \(f(x) = x^2\) and \(f(x) = \cos(x)\) are even functions. For any even function, substituting \(-x\) into the equation will give you the same result as the original function.

Odd Functions

An odd function has rotational symmetry around the origin. This means that if you rotate the graph 180 degrees around the origin, it looks the same. An odd function satisfies the equation:
\[ g(-x) = -g(x) \]

This property indicates that the value of the function at a positive x-value is the negative of the value at the corresponding negative x-value. For example, the functions \(g(x) = x^3\) and \(g(x) = \sin(x)\) are odd functions. For any odd function, substituting \(-x\) into the equation will give you the negative of the original function.

Product of Functions

When multiplying functions, the parity (evenness or oddness) of the result depends on the parities of the functions being multiplied. Let's consider the combinations:

• Even times Even: If \(f(x)\) and \(h(x)\) are both even, then \( (f(x) \times h(x))(-x) = f(-x) \times h(-x) = f(x) \times h(x)\). The product is even.
• Odd times Odd: If \(g(x)\) and \(k(x)\) are both odd, then \( (g(x) \times k(x))(-x) = g(-x) \times k(-x) = (-g(x)) \times (-k(x)) = g(x) \times k(x)\). The product is even.
• Even times Odd: If \(f(x)\) is even and \(g(x)\) is odd, then \( (f(x) \times g(x))(-x) = f(-x) \times g(-x) = f(x) \times (-g(x)) = -(f(x) \times g(x))\). The product is odd.

Derivative of Functions

The derivative of a function changes its parity based on the original function's parity:

• Derivative of an Even Function: If \(f(x)\) is an even function, then:
  \[ f(-x) = f(x) \]
  Differentiating both sides with respect to x, we get:
  \[ \frac{d}{dx}[f(-x)] = \frac{d}{dx}[f(x)] \]
  This simplifies to:
  \[ -f'(-x) = f'(x) \]
  This shows that \(f'(x)\) is an odd function.
• Derivative of an Odd Function: If \(g(x)\) is an odd function, then:
  \[ g(-x) = -g(x) \]
  Differentiating both sides with respect to x, we get:
  \[ \frac{d}{dx}[g(-x)] = \frac{d}{dx}[-g(x)] \]
  This simplifies to:
  \[ -g'(-x) = -g'(x) \]
  Therefore,
  \[ g'(-x) = g'(x) \]
  This shows that \(g'(x)\) is an even function.