In mathematics, finding the product of two functions is a common task that arises in various branches such as algebra, calculus, and applied mathematics. The operation involves multiplying two functions together, which can be more complex than multiplying mere numbers due to the dependency on variables.

This article will guide you through the step-by-step process of how to find the product of two given functions and elaborate on some of their applications with examples to illustrate the process.

**Understanding Function Multiplication**

Before we multiply two functions, itâ€™s important to understand what a function represents in mathematics. A function is a relation or a rule that assigns each input exactly one output. When we talk about the product of two functions, we are essentially describing a new function that represents the multiplication of the outputs of the two individual functions for each input in their domain.

**Step-by-Step Guide to Finding A Product Functions**

**Step 1: Identify the Functions**

Assume we have two functions, f(x) and g(x). For example, let’s consider the following:

f(x) = x + 3g(x) = 2x – 5

**Step 2: Write Down the Multiplication**

Write the multiplication explicitly. When we multiply f(x) by g(x), it is denoted as:

(f*g)(x) = f(x) * g(x)

It’s important to note that when multiplying functions, we must distribute each term in the first function by each term in the second function.

**Step 3: Execute the Multiplication**

Now, perform the multiplication for each term: (f*g)(x) = (x + 3)(2x – 5)

This results in: (f*g)(x) = x*(2x) + x*(-5) + 3*(2x) + 3*(-5)

Which simplifies to: (f*g)(x) = 2x^{2} – 5x + 6x – 15

Combine like terms: (f*g)(x) = 2x^{2} + x – 15

So the product of f(x) and g(x) is a new function h(x) = 2x^{2} + x – 15.

**Applications of Function Products**

Multiplying functions can have applications in various mathematical contexts such as area calculations, volume integrals in calculus, and the formation of more complex functions in algebra.

**Calculating Areas**

In geometry, the product of two functions can represent the area of a rectangle. If f(x) represents the length and g(x) the width, then the product h(x) = (f*g)(x) gives the area of the rectangle at any point x.

**Volume Integrals**

In calculus, when rotating a region around an axis, the product of functions can be used to set up an integral for calculating volumes. This is often used in the disk and washer methods for volume calculations.

**Combining Properties**

In physics and engineering, the product of functions is useful when combining properties such as speed and time to compute distance or voltage and current to determine electrical power.

**Special Considerations**

While finding the product of functions generally follows the multiplication rules learned in elementary algebra, there are some special cases.

**Non-Overlapping Domains:**If the functions have non-overlapping domains, the product is undefined for inputs outside of the intersection of the domains.**Zero Functions:**If either function is the zero function, the product will also be the zero function regardless of the other function.

**Conclusion**

The product of two functions extends the concept of simple multiplication to more complex scenarios involving variable relationships. Through careful execution of known algebraic rules, one can find the product that itself is another function encapsulating the combined effect of the original functions. Whether in pure mathematics or applied sciences, the ability to find the product of two functions is an invaluable skill in both theoretical and practical endeavors.