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Chapter 4: Problem 5

Fill in the blank. The function \(f(x)=a^{x}\) is ________ if \(a>1\) and ______ if \(0

Short Answer

Expert verified
The function is increasing if a > 1 and decreasing if 0 < a < 1.

Step by step solution

01

- Understanding the function type

The given function is an exponential function of the form f(x) = a^x. Here, the base 'a' of the exponent determines the behavior of the function.
02

- Analyze the case when a > 1

When the base 'a' is greater than 1 (i.e., a > 1), the function is increasing. This is because as x increases, the value of a^x also increases.
03

- Analyze the case when 0 < a < 1

When the base 'a' is between 0 and 1 (i.e., 0 < a < 1), the function is decreasing. This is because as x increases, the value of a^x decreases.

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Key Concepts

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

Increasing Functions
An increasing function is one where the function's value rises as the input (or x-value) increases. Put simply, as you move to the right on the x-axis, the graph of the function goes up. For exponential functions like \( f(x) = a^x \), if the base 'a' is greater than 1, the function is increasing. This means that the function grows faster and faster as x gets larger.

When a function is increasing, it is useful in scenarios where growth is exponential, such as in population growth, compound interest, or bacteria reproduction.

In practical terms:
  • If \( a > 1 \), the function \( f(x) = a^x \) increases as x increases.
Understanding how increasing functions work helps you predict and model real-world behaviors effectively.
Decreasing Functions
A decreasing function is one where the function's value drops as the input (or x-value) increases. Simply put, as you move to the right on the x-axis, the graph of the function goes down. For exponential functions like \( f(x) = a^x \), if the base 'a' is between 0 and 1, the function is decreasing. This means that the function's value gets smaller and smaller as x gets larger.

Decreasing functions are useful in scenarios where measures are diminishing, such as radioactive decay or cooling of an object.

In practical terms:
  • If \( 0 < a < 1 \), the function \( f(x) = a^x \) decreases as x increases.
Knowing how decreasing functions behave can help you understand and model processes that involve decay or reduction.
Base of an Exponent
The base of an exponential function \( f(x) = a^x \) plays a crucial role in determining whether the function is increasing or decreasing.

Here are the key points about the base of an exponent:
  • If \( a > 1 \), the function is increasing. Examples include \( 2^x \), \( 3^x \), and \( 10^x \).
  • If \( 0 < a < 1 \), the function is decreasing. Examples include \( (1/2)^x \), \( (1/3)^x \), and \( (1/10)^x \).
Understanding this concept helps you predict the behavior of the function just by looking at its base.
Knowing whether an exponential function will increase or decrease can aid you in various fields like finance, biology, and physics. It also simplifies solving problems related to growth and decay.

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Most popular questions from this chapter

Present Value of a \(\mathrm{CD}\) What amount (present value) must be deposited today in a certificate of deposit so that the investment will grow to \(\$ 20,000\) in 18 years at \(6 \%\) compounded continuously.

Solve \(2 x-5=7 x\)

Computers per Capita The number of personal computers per 1000 people in the United States from 1990 through 2010 is given in the accompanying table (Consumer Industry Almanac, www.c-i-a.com). a. Use exponential regression on a graphing calculator to find the best- fitting curve of the form \(y=a \cdot b^{x},\) where \(x=0\) corresponds to 1990. b. Write your equation in the form \(y=a e^{e x}.\) c. Assuming that the number of computers per 1000 people is growing continuously, what is the annual percentage rate? d. In what year will the number of computers per 1000 people reach \(1500 ?\) e. Judging from the graph of the data and the curve, does the exponential model look like a good model? $$\begin{array}{|l|c|} \hline \text { Year } & \begin{array}{c} \text { Computers } \\ \text { per 1000 } \end{array} \\ \hline 1990 & 192 \\ 1995 & 321 \\ 2000 & 628 \\ 2005 & 778 \\ 2010 & 932 \\ \hline \end{array}$$

Find the domain and range of the function $$f(x)=-\frac{1}{2} 3^{x-5}+7$$.

Which exponential and logarithmic functions are increasing? Decreasing? Is the inverse of an increasing function increasing or decreasing? Is the inverse of a decreasing function increasing or decreasing? Explain.
See all solutions

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