91Ó°ÊÓ

In mathematics, a function is called decreasing if, as the input (or x-value) increases, the output (or y-value) decreases. For exponential functions like the one given in the exercise, whether the function is increasing or decreasing depends on the value of the base.

If the base of an exponential function is between 0 and 1, the function is decreasing. This means that as the x-value gets larger, the corresponding y-value gets smaller. For example, in the function given by the exercise, \(f(x)=\left( \frac{3}{5} \right)^x\), we have a base of 3/5, which is between 0 and 1. Therefore, this function is decreasing. Decreasing functions can be useful in various contexts, such as modeling decay or reduction over time.

The base of an exponential function is critical in determining the function's behavior. An exponential function can be written in the form \(f(x)=a^x\). Here, 'a' is the base, and 'x' is the exponent. The base determines whether the function increases or decreases as x increases.

If the base is greater than 1, the function is increasing. This means that the y-value gets larger as the x-value gets larger. Conversely, if the base is between 0 and 1, the function is decreasing. This is the scenario in our exercise, where the base \(\left( \frac{3}{5} \right)\) is between 0 and 1.

Here's a quick summary:

• \texttt{a > 1}: Function is increasing
• \texttt{0 < a < 1}: Function is decreasing

Recognizing the base helps you quickly understand how the function behaves without having to plot it.

Analyzing the behavior of exponential functions involves understanding how the function changes as x varies. The most straightforward way is to look at the base.

In our specific example of \(f(x)=\left( \frac{3}{5} \right)^x\), we see:
1. **Base Analysis:** Since 3/5 is between 0 and 1, the function will decrease over its domain.
2. **End Behavior:** As x tends to positive infinity, the values of \(f(x)\) tend towards 0. This means as you go further to the right on the x-axis, the y-values shrink and get very close to 0.
3. **Graph Characteristics:** The graph of a decreasing exponential function will slope downward from left to right. No matter where you start, you will see the y-values getting smaller.

By understanding these key points about function behavior analysis, you can predict how an exponential function will behave. This not only helps with solving problems but also gives insight into real-world situations modeled by such functions, like radioactive decay or cooling substances.