Problem 5 - Precalculus Functions and Graphs

Question

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

Step-by-Step Solution

Answer

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

Step 1 - 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.

Step 2 - 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.

Step 3 - 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.

Key Concepts

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.