Problem 109 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 2} x$$

Step-by-Step Solution

Answer

The function \(f(x) = \log_{1/2}x\) is decreasing on its domain.

Step 1: Rewrite the logarithmic function in exponential form

To understand the behavior of the logarithmic function, rewrite it in its equivalent exponential form. Since the function is given by \(f(x) = \log_{1/2}x\), its exponential form is \( (1/2)^{y} = x \). This tells us that when \(x\) is less than 1, \(y\) is positive; and when \(x\) is greater than 1, \(y\) is negative.

Step 2: Understand the logarithmic base in function

In the function \(f(x) = \log_{1/2}x\), the base of the logarithm is \(\frac{1}{2}\), which is a positive number less than 1. For logarithms, if the base is less than 1, the function will decrease as \(x\) increases, because as \(x\) becomes larger, the value required to raise the base to reach \(x\) becomes smaller.

Step 3: Identify the domain of the function

The domain of \(f(x) = \log_{1/2}x\) is determined by the fact that logarithms are only defined for positive real numbers. Thus, \(x > 0\).

Step 4: Analyze the trend of the function

Given that the base of the logarithm is \(\frac{1}{2}\), which is less than 1, the function \(f(x) = \log_{1/2}x\) is decreasing across its domain. This means that as \(x\) increases, the function output decreases.

Key Concepts

Exponential Form

Logarithmic functions can be challenging at first glance, but breaking them down into their exponential form can make things clearer. If you're given a logarithmic function, like \(f(x) = \log_{1/2}x\), you can convert it into an exponential form to better understand it. By converting, you reveal the relationship: \((1/2)^{y} = x\). This means if you can determine \(x\), you can find \(y\) by asking what power was needed to make \(x\) from the base \(1/2\).
This conversion is helpful because exponential equations are sometimes easier to visualize and understand. They illustrate how changes in \(y\) relate to changes in \(x\). Numbers less than 1 in the base, like \(1/2\), flip the usual growth pattern seen in bases greater than 1.

Function Behavior

With logarithmic functions, the base of the logarithm plays a crucial role in determining how the function behaves. In the function \(f(x) = \log_{1/2}x\), the base is \(1/2\). Since this is a number between 0 and 1, it significantly affects the function's behavior.

• If the base is more than 1, the function will increase with \(x\). This is because larger values of \(x\) need larger powers to be reached.
• For bases less than 1, such as \(1/2\), the function decreases as \(x\) increases. So, as \(x\) goes up, the live function value goes down.

In our specific case of \(\log_{1/2}x\), as \(x\) increases, \(y\), or the output of the function, gradually decreases. This behavior is opposite to what might be expected with a common base greater than 1.

Domain of Functions

Knowing the domain of logarithmic functions is essential for understanding where these functions can be used. For any logarithmic function, the argument \(x\) must be positive. This stems from the fact that you can't take the logarithm of zero or negative numbers.
In the function \(f(x) = \log_{1/2}x\), the rule is no different. The domain of this function is \(x > 0\). Anyone analyzing such a function must start by recognizing that \(x\) cannot be zero or negative.

• Logarithmic functions always require positive inputs.
• Make sure to identify clearly where \(x\) values are valid when interpreting or graphing.

This makes the function applicable only within the positive real number range. Understanding the domain helps in graphing the function and checking its continuity.