Problem 57 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. An exponential function can be decreasing.

Step-by-Step Solution

Answer

True, an exponential function can be decreasing if the base is between 0 and 1.

Step 1: Define Exponential Function

An exponential function is of the form \( f(x) = a \cdot b^x \), where \( a eq 0 \) and \( b > 0 \). Here, \( a \) is a constant, and \( b \) is the base.

Step 2: Understand Base Condition

For an exponential function, if \( 0 < b < 1 \), the function decreases as \( x o \-\infty \). This means the function values become smaller as \( x \) increases.

Step 3: Analyze Decreasing Condition

When \( 0 < b < 1 \), the multiplication by \( b \) will continuously reduce the value of \( b^x \) with increasing \( x \). Thus, such an exponential function is decreasing over its entire domain.

Step 4: Conclusion

Based on the analysis, an exponential function can indeed be decreasing if the base \( b \) is between 0 and 1. Therefore, the statement is true.

Key Concepts

Decreasing Functions

An exponential function generally takes the form of \( f(x) = a \cdot b^x \), where \( a eq 0 \) and \( b > 0 \). While we often associate exponential functions with growth, they can indeed be decreasing. This happens under the condition when the base \( b \) is between 0 and 1.

• For instance, if \( b = 0.5 \), multiplying by 0.5 reduces the preceding value as \( x \) increases.
• As \( x \) approaches infinity, \( b^x \) becomes smaller and smaller, approaching zero.

Therefore, when the base \( b \) is in this range, the exponential function will continuously decrease, reflecting a reduction rather than growth as \( x \) increases. This ability to decrease is what differentiates the exponential function from other commonly known functions like linear or quadratic functions, which don't exhibit this kind of behavior in their negative domain.

Base Condition

One of the crucial aspects influencing the behavior of an exponential function is its base condition. The base \( b \) in an exponential function \( f(x) = a \cdot b^x \) must be greater than zero for the function to be properly defined. However, the real magic happens with \( 0 < b < 1 \).

• This range implies that with every increment in \( x \), the magnitude of \( b^x \) decreases.
• Visualize this as a percentage reduction with each step forward, akin to a discount factor.

Essentially, the smaller the base (below 1 but above zero), the more sharply the function values decline. Understanding this base condition is key to predicting and explaining the behavior of decreasing exponential functions, allowing precise modeling of relationships like depreciation or radioactive decay.

Function Behavior

The behavior of an exponential function heavily relies on its base and exponent. When considering \( f(x) = a \cdot b^x \), varying the base \( b \) adjusts whether the function grows or shrinks.

• If \( b > 1 \), the function exhibits growth—every increase in \( x \) yields a larger \( f(x) \).
• Conversely, if \( 0 < b < 1 \), it showcases a decreasing trend, meaning every unit increase in \( x \) leads to a smaller \( f(x) \).

Furthermore, it’s interesting that the constant \( a \) mainly scales the function, shifting it up or down without altering the growth or decay behavior. Therefore, by understanding these foundational principles of behavior, you can forecast and manipulate exponential functions to model various real-life phenomena like population changes, financial investments, or cooling processes—all encapsulated under the powerful umbrella of exponential behavior.