Problem 32 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that the product of two even or two odd functions is an even function, whereas the product of an even and an odd function is an odd function.

Step-by-Step Solution

Answer

To summarize, the product of two even functions or two odd functions is always an even function, since \(h(-x) = h(x)\) in both cases. On the other hand, the product of an even and an odd function is always an odd function, since \(h(-x) = -h(x)\).

Step 1: Definition of even and odd functions

A function \(f(x)\) is called an even function if it satisfies the condition \(f(-x) = f(x)\) for all x in its domain. A function \(g(x)\) is called an odd function if it satisfies the condition \(g(-x) = -g(x)\) for all x in its domain.

Step 2: 1. Product of two even functions

Let \(f(x)\) and \(g(x)\) be two even functions. We want to prove that their product, \(h(x) = f(x)g(x)\), is also an even function. To prove this, we need to show that \(h(-x) = h(x)\) for all x in its domain. \(h(-x) = f(-x)g(-x)\) since both f and g are even functions, we have: \(f(-x) = f(x)\) and \(g(-x) = g(x)\) So, \(h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)\) Thus, the product of two even functions is an even function.

Step 3: 2. Product of two odd functions

Let \(f(x)\) and \(g(x)\) be two odd functions. We want to prove that their product, \(h(x) = f(x)g(x)\), is an even function. To prove this, we need to show that \(h(-x) = h(x)\) for all x in its domain. \(h(-x) = f(-x)g(-x)\) since both f and g are odd functions, we have: \(f(-x) = -f(x)\) and \(g(-x) = -g(x)\) So, \(h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x)\) Thus, the product of two odd functions is an even function.

Step 4: 3. Product of an even and an odd function

Let \(f(x)\) be an even function and \(g(x)\) be an odd function. We want to prove that their product, \(h(x) = f(x)g(x)\), is an odd function. To prove this, we need to show that \(h(-x) = -h(x)\) for all x in its domain. \(h(-x) = f(-x)g(-x)\) since f is an even function and g is an odd function, we have: \(f(-x) = f(x)\) and \(g(-x) = -g(x)\) So, \(h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -(f(x)g(x)) = -h(x)\) Thus, the product of an even and an odd function is an odd function.

Key Concepts

Product of Functions

When you multiply two functions, you are creating what is known as the "product of functions." This is an important concept in mathematics and is especially interesting when examining the types of functions in the product.

For two functions, say \(f(x)\) and \(g(x)\), the product is simply \(h(x) = f(x) \cdot g(x)\). Depending on whether \(f(x)\) and \(g(x)\) are even or odd, the resulting function \(h(x)\) can behave differently.

• If both functions are even, their product is even.
• If both functions are odd, their product is also even.
• If one function is even and the other is odd, the product is odd.

These outcomes rely on the specific properties of even and odd functions, which we will delve into further.

Even Function

An even function is one of the fundamental concepts within the realm of functions. Even functions have a distinctive symmetry about the y-axis. This symmetry is expressed mathematically by the condition \(f(-x) = f(x)\) for all x in the function's domain.

• Graphically, even functions mirror across the y-axis.
• This mirroring implies that plugging in \(-x\) gives the same result as plugging in \(x\).

When even functions are multiplied together, this symmetry is preserved, making their product also an even function. This is because the replacement \((f(-x) = f(x))\) yields no change when both functions are themselves even. So, for two even functions \(f(x)\) and \(g(x)\), their product \(h(x) = f(x)g(x)\) maintains evenness, confirming \(h(-x) = h(x)\).

Odd Function

Odd functions offer a charming alternate symmetry. Their hallmark feature is rotational symmetry about the origin, which mathematically means \(f(-x) = -f(x)\). You can visualize this by imagining a 180-degree rotation about the origin, where the function appears unchanged.

• A typical characteristic of an odd function is that they intercept the origin, though this is not a strict requirement.
• Rotational symmetry involves flipping both horizontally and vertically, creating this interesting property.

When you multiply two odd functions together, interestingly, they form an even function. This occurs because the negatives cancel each other out, leading to the condition \(h(-x) = f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x) = h(x)\). On the other hand, mixing an odd function \(g(x)\) with an even function \(f(x)\) results in an odd product, as \(h(-x) = f(-x)g(-x) = f(x)(-g(x)) = -h(x)\). This interplay of symmetry types provides enriching insights into the behavior of functions.