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Exponential growth occurs when the value of a function increases rapidly as the input increases. This type of growth is often represented by functions of the form \( f(x) = a \cdot b^x \), where \( a \) is a positive constant, and \( b \) is the base of the exponential function. The key feature of exponential growth is that the base \( b \) is greater than 1.
Some common instances of exponential growth include:

• Population growth, where the number of individuals increases exponentially over time.
• Compound interest, where the amount of money increases at an exponential rate.
• Spread of viral infections, where the number of infected individuals grows rapidly.

Exponential functions with growth can be identified quickly. If the base of the exponent \( b > 1 \), expect rapid increases. This growth is proportional to the current value, meaning as the function values grow, they do so by increasing factors rather than adding numbers. Keep in mind that the initial value \( a \) plays a role in determining the starting point of the growth. However, the base primarily dictates the growth rate.

Exponential decay describes the process where the value of a function decreases rapidly over time. Such processes are characterized by functions of the form \( f(x) = a \cdot b^x \), where \( 0 < b < 1 \). Here, the base \( b \) is a fraction less than 1, leading to a decrease in the function values as \( x \) increases.
Examples of exponential decay include:

• Radioactive decay, where substances lose their activity exponentially over time.
• Depletion of a resource, such as a battery drainage.
• Cooling of hot objects to room temperature.

In the context of the provided function \( f(x) = 2^{-x} \), note how the negative exponent transforms the expression into one that resembles an exponential decay situation. Rewriting \( 2^{-x} = \left( \frac{1}{2} \right)^x \) shows the fraction form \( \frac{1}{2} \) with a base between 0 and 1, confirming the decay nature. As \( x \) increases, the value of the function \( f(x) \) approaches zero, showcasing the characteristic exponential decline.

A negative exponent in an expression, such as \( a^{-x} \), signifies the reciprocal of the base raised to the corresponding positive exponent. This can be written as \( \frac{1}{a^x} \). The role of the negative exponent greatly influences how the function behaves and often relates to exponential decay.
Understanding negative exponents involves recognizing:

• Inverse operations, where multiplication becomes division. For example, \( 2^{-x} \) is the same as \( \frac{1}{2^x} \).
• The transformation of expressions: Changing \( b^{-x} \) into \( \left(\frac{1}{b}\right)^x \) can help identify cases of exponential decay.
• Simplifying expressions: Often used to rewrite terms for easier handling of equations.

Analyzing the function \( f(x) = 2^{-x} \) from the original exercise, the negative exponent indicates a reciprocal relationship with the base, resulting in outcomes less than 1 as \( x \) increases. This transformation turns growth potential into a decay scenario, affirming the power of negative exponents in altering outcomes and function behavior.