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Chapter 9: Problem 27

Determine whether each function represents exponential growth or decay. $$ y=0.2(5)^{-x} $$

Short Answer

Expert verified
The function represents exponential decay.

Step by step solution

01

Identify the Base of the Exponential Function

In the function \( y = 0.2(5)^{-x} \), identify the base. The base is the number being raised to the power, which in this case is \( 5^{-1} \). Express this as \( \left( \frac{1}{5} \right)^x \).
02

Determine the Exponent

The exponent in \( y = 0.2(5)^{-x} \) is \(-x\). This means the base is raised to the power of the negative of \( x \).
03

Assess the Effect of the Negative Exponent

A negative exponent on a base greater than one, like \( \frac{1}{5} \) in \( \left( \frac{1}{5} \right)^x \), indicates division, leading to exponential decay because as \( x \) increases, the value of the base fraction raised to a positive power decreases.
04

Conclusion

Since the base here, \( \frac{1}{5} \) (after simplifying \( 5^{-1} \)) is less than 1, and is raised to a positive \( x \) when rewritten, the function decreases as \( x \) increases, signifying exponential decay.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
An exponential function is a mathematical expression where a constant base is raised to a variable exponent. These functions can rapidly increase or decrease, making them unique in their mathematical behavior. In most cases, they take the form \( y = a \,b^x \), where:
  • \( a \) is a constant that represents the starting value or vertical shift.
  • \( b \) is the base of the power.
  • \( x \) is the exponent, which is usually a variable.
When the base \( b \) is greater than 1, the function typically models exponential growth. Conversely, if the base is between 0 and 1, it often shows exponential decay. This is because the power increases the base's effect, either expanding toward infinity or shrinking toward zero.
Negative Exponents
Negative exponents can be a bit tricky at first, but they are straightforward once you understand the concept. A negative exponent means that the reciprocal of the base will be taken. This can be expressed with the rule: \( a^{-n} = \frac{1}{a^n} \). The concept of negative exponents is useful to transform expressions and understand their growth or decay behavior.
Let's look at an application: In the expression \( 5^{-x} \), the negative exponent indicates that the base 5 is in the denominator when rewritten as \( \frac{1}{5^x} \). Thus, as \( x \) increases, the value of \( \frac{1}{5^x} \) decreases, demonstrating the concept of exponential decay.
Base Identification
Identifying the base in an exponential function is the cornerstone to understanding its behavior. In an expression like \( y = 0.2(5)^{-x} \), the base is the number attached directly to the exponent \( x \). Here, it is crucial to recognize the form by simplifying \( 5^{-x} \) to \( (\frac{1}{5})^x \). This adjustment helps us clearly understand that the base is effectively \( \frac{1}{5} \) due to the negative exponent.
Knowing whether the base is greater than 1 or less than 1 gives insight into whether the function depicts growth or decay. In our function, since \( \frac{1}{5} \) is less than 1, the function illustrates decay.
Function Behavior Analysis
Analyzing the behavior of a function boils down to a few simple steps to determine how the function changes as the variable increases. Begin by observing the base: in our function \( y = 0.2(5)^{-x} \), converting the base to \( \frac{1}{5} \) helps assess its effects.
  • Since \( \frac{1}{5} \) is less than 1, you anticipate the function symbolizes decay, suggesting it decreases as \( x \) increases.
  • The presence of negative exponents typically enhances decay, as it inverts the base to its fractional form.
Conclusively, examining both the base and the exponent's sign together underlines the decisively decaying pattern, as seen when the input \( \frac{1}{5} \) repeatedly multiplies in smaller fractions for each unit increase in \( x \).

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