Problem 69 - Precalculus : Building Concepts and Connections

Question

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(t)=\log _{1 / 3} t$$

Step-by-Step Solution

Answer

The domain of the function \(f(t)=\log _{1 / 3} t\) is \(t > 0\). When graphing the function, there is a vertical asymptote at \(t=0\).

Step 1: Finding the Domain

The domain of \(f(t)=\log _{1 / 3} t\) is found by setting \(t > 0\). This is because logarithms are only defined for positive numbers.

Step 2: Drawing the Graph

First, sketch the y-axis and x-axis. For \(f(t)=\log _{1 / 3} t\), as the base of the logarithm is less than 1, the graph of the function will show a declining curve from the right, as this is the property of the graph of a logarithmic function whose base is between 0 and 1.

Step 3: Identifying Asymptotes

In the graph, there will be a vertical asymptote at \(t=0\). This can be drawn as a dotted vertical line on the x-axis - this is also a part of the domain which shows that the function is not defined at \(t=0\). The function never touches this line and tends to negative infinity as \(t\) approaches 0 from the right.

Key Concepts

Domain of a Function

The domain of a function refers to the set of input values (typically represented by 'x' or in this case 't') for which the function is defined. Understanding the domain is crucial because it tells us the range of values we can plug into the function without running into issues like undefined terms or negative square roots in the case of square root functions.

For logarithmic functions like the one in our exercise, the domain is all positive real numbers, because the logarithm of a negative number or zero is not defined in the real number system. Specifically, for the function f(t) = \(\log_{1/3} t\), the input 't' must be greater than zero, that is, t > 0. This exclusion of zero and negative numbers is fundamental to all logarithmic functions, and it's key to grasp this concept to avoid errors in calculating and graphing.

Graphing Logarithmic Functions

To graph a logarithmic function, one should start by identifying its key features such as the domain, range, intercepts, and asymptotes. For the function f(t) = \(\log_{1/3} t\), we begin by sketching the axes and noting that 't' must be positive. We then consider the base of the logarithmic function. When the base is between 0 and 1, as in this case where the base is \(1/3\), the function will display a decreasing curve. This is because as 't' increases, the value of \(\log_{1/3} t\) actually decreases, due to the inverse nature of logarithmic functions. Remember, when graphing, to plot a few key points by choosing convenient values for 't' and determining their corresponding 'f(t)' values, connecting these points smoothly to reveal the curve of the function. These steps provide a visual representation that enhances understanding.

Vertical Asymptotes

A vertical asymptote is a line that a function approaches but never actually reaches or crosses. This behavior highlights the boundary of a function's domain and often signals where the function is undefined. In the context of logarithmic functions, a vertical asymptote will occur at 't=0'.

Graphically, the graph tends to infinity as it approaches the asymptote from the right-hand side for decreasing functions like our \(\log_{1/3} t\). It's vital to illustrate this asymptote as a dashed vertical line on the graph to signify that the function will not have any value at 't=0'. To observe the asymptotic behavior, one can plug values of 't' increasingly close to zero into the function and notice how the output values become more and more negative, reinforcing the concept that the function is undefined at 't=0'. This understanding is essential for accurate graphing and analysis of the behavior of logarithmic functions.