Problem 110 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\). Is \(f\) increasing or decreasing on its domain? $$f(x)=\log _{1 / 3} x$$

Step-by-Step Solution

Answer

The function is decreasing on its domain.

Step 1: Determine Base of the Logarithm

The function given is \( f(x) = \log_{1/3} x \). Here, \( a = \frac{1}{3} \) is the base of the logarithm. Since the base is \( 1/3 < 1 \), this is a decreasing logarithmic function.

Step 2: Domain of Function

The domain of \( f(x) = \log_{1/3} x \) is \( x > 0 \), since logarithm functions are only defined for positive values of \( x \).

Step 3: Increasing or Decreasing

Since the base of the logarithm is less than 1, the function \( f(x) = \log_{1/3} x \) decreases as \( x \) increases. Therefore, the graph of the function will slope downwards from left to right.

Step 4: Conclusion about the Behavior

Based on the analysis, \( f(x) = \log_{1/3} x \) is a decreasing function on its domain \( x > 0 \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values that the function can accept. In simpler terms, it is the set of all possible values of the variable for which the function is defined. Logarithmic functions, in particular, have specific domain restrictions. For the function \( f(x) = \log_{1/3} x \), the domain is determined by the properties of logarithms. Logarithms are only defined for positive real numbers. This means that in the given function, the value of \( x \) must be greater than zero.

Key points to remember about the domain of logarithmic functions:

• Logarithms cannot take negative numbers or zero as input, only positive numbers.
• The domain for \( \log_b x \) where \( b \) is the base, is \( x > 0 \).
• Understanding the domain is crucial before graphing or analyzing the function's behavior.

Decreasing Function

A function is termed as decreasing when as the input value, commonly denoted as \( x \), increases, the output value, or \( f(x) \), decreases. In the context of the logarithmic function \( f(x) = \log_{1/3} x \), it's considered decreasing across its domain. This happens because the base of the logarithm, \( 1/3 \), is less than 1.

Here's why a base less than 1 leads to a decreasing function:

• The rule of logarithms dictates that when the base \( b \) satisfies \( 0 < b < 1 \), the function decreases.
• Thus, higher values of \( x \) result in smaller values of \( f(x) \).
• The decrease occurs smoothly, creating a downward trend from left to right on the graph.

Therefore, for \( f(x) = \log_{1/3} x \), as \( x \) progresses from small to large within its domain of \( x > 0 \), \( f(x) \) declines.

Graphing Functions

Graphing a function involves plotting its points on a coordinate plane to visualize its behavior. When graphing \( f(x) = \log_{1/3} x \), understanding its properties helps in sketching it accurately.

Here's how you can approach graphing this function:

• Start by noting that the domain is \( x > 0 \), meaning the graph starts just to the right of the y-axis.
• Recognize that the function is decreasing, depicted by a curve moving downward as you move from left to right.
• The graph approaches but never touches the y-axis, aligning with the fact that the logarithm is undefined at \( x = 0 \).

These characteristics help ensure your graph is accurate. By knowing the nature of the function, its base, and its domain, you can confidently plot the points and draw the curve that represents the behavior of \( f(x) = \log_{1/3} x \).