91Ó°ÊÓ

When we talk about the product of functions, we're considering the multiplication of two different functions to create a third new function. If you have two functions, say \( f(x) \) and \( g(x) \), their product is \( h(x) = f(x) \cdot g(x) \). This means for each value of \( x \), \( h(x) \) is the result of multiplying the corresponding outputs of \( f(x) \) and \( g(x) \).

This new function \( h(x) \) derives its characteristics from both \( f \) and \( g \). Whether it increases, decreases, or remains constant can depend on various factors, such as the individual behavior of \( f(x) \) and \( g(x) \).

One might expect that multiplying functions with similar behaviors (e.g., both increasing) might give a predictable result, but this is not always the case. The sign and magnitude of the functions play a crucial role, especially when negative values are involved.

Negative values in a function can strongly influence the way a product function behaves. Suppose both \( f(x) \) and \( g(x) \) are increasing functions. It seems intuitive to think their product \( h(x) = f(x)g(x) \) would also be increasing.

However, this changes if one or both of these functions take on negative values. Multiplying a positive number with a negative one results in a negative number. Similarly, multiplying two negative numbers results in a positive number. This switch in sign can make a product function behave differently than expected.

For instance, let's consider \( f(x) = x \) and \( g(x) = -x \) over the interval \([-1, 1]\). Individually, both functions are increasing across this range. However, their product, \( h(x) = -x^2 \), behaves differently. Instead of increasing, \( h(x) \) actually decreases, demonstrating the powerful impact of negative values.

Non-negative functions are those which do not take on any negative values; they are always zero or positive. When we consider two such functions, both being non-negative and increasing, their product is often easier to predict.

With both functions remaining on the positive side of the number line, the product function will indeed be increasing. Why? Because multiplying two positives gives a positive, and their increasing nature contributes to the overall increase in the product function.

For example, if \( f(x) = x + 1 \) and \( g(x) = x + 2 \) over a range where \( x > 0 \), both are non-negative and increase as \( x \) increases. Their product, \( h(x) = (x + 1)(x + 2) \), increases smoothly without any complicating sign changes.

This illustrates that non-negative functions maintain a consistent growth pattern when multiplied, highlighting the importance of non-negative values when analyzing the behavior of product functions.