Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3} x+1$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \log_{3}x + 1\) is \(x > 0\), the vertical asymptote is at \(x = 0\), and the x-intercept is at \(x = \frac{1}{3}\)

Step 1: Finding the Domain

The domain of a logarithmic function is the set of x-values that would give a valid output. Since logging negative values and zero are undefined in the real number system, the domain in this case would be \(x > 0\)

Step 2: Finding the x-Intercept

The x-intercept is the point where the graph cuts the x-axis. To find this, set \(f(x)\) equal to zero and solve for \(x\). Here this gives:\[0 = \log_{3}x + 1\]Subtract 1 from both sides:\[-1 = \log_{3}x\]Then change to exponential form:\[3^{-1} = x\]Which simplifies to \(x = \frac{1}{3}\]. Thus, the x-intercept is \(\frac{1}{3}\)

Step 3: Finding Vertical Asymptote

Vertical asymptote is a vertical line that the function approaches but never crosses or touches. For the function \(f(x) = \log_{3}x + 1\), the vertical asymptote is x = 0 because as x approaches zero from the right, \(f(x)\) tends to negative infinity.

Step 4: Graphing the function

Using a graphing utility (like a graphing calculator or software), the function \(f(x) = \log_{3}x + 1\) can be plotted. Set the range of x as close to zero as possible and to a number large enough to visualize the important features of the graph.

Key Concepts

Graphing Utility

A graphing utility is a tool used to plot mathematical functions and visualize their behavior. This tool can be a physical calculator with graphing capabilities or software on a computer or mobile device. It helps to display intricate functions like logarithmic curves of the function\[ f(x) = \log_{3}x + 1 \] Effectively graphing this function allows one to understand its shape, determine intercepts, asymptotes, and evaluate behavior over different intervals.
Using a graphing utility effectively involves the following:

• Input the function correctly.
• Adjust the window settings to ensure that all important features of the graph, including the axis intersections and asymptotes, are visible.
• Analyze the graph to interpret the behavior of the function, such as how it increases as \(x\) increases.

Graphing utilities provide a big-picture view of the function's performance across numerous inputs, hence aiding in understanding concepts such as domain, asymptotes, and intercepts.

Domain

The domain of a logarithmic function is crucial, as it defines the valid input values for \( x \). For the function \( f(x) = \log_{3}x + 1 \), it's important to know that logarithms are defined only for positive numbers. Consequently, the domain is:\[ x > 0 \]This means that any value of \( x \) must be greater than zero for the logarithm to provide a real number. If \( x \) were zero or negative, the function would be undefined.
Knowing the domain helps in understanding where the function exists mathematically and visually on its graph. Valid domain values result in a curve that continues infinitely towards the vertical asymptote and beyond.

Vertical Asymptote

A vertical asymptote is a straight line that indicates where a function's value grows considerably large in absolute terms but never actually meets or crosses. For the logarithmic function \( f(x) = \log_{3}x + 1 \), the vertical asymptote occurs at:\[ x = 0 \]This is because, as \( x \) approaches zero from the positive side, the logarithmic output descends towards negative infinity. However, the function does not exist for \( x \leq 0 \) which emphasizes the limit at this asymptote.
Understanding asymptotes is essential as they mark the boundaries where significant changes in the function's direction occur. They play a vital role in creating an accurate graphical representation of the function.

X-intercept

The x-intercept of a function is the point at which the graph crosses the x-axis. It's where the output of the function equals zero. For the logarithmic function \( f(x) = \log_{3}x + 1 \), finding the x-intercept involves setting the equation to zero:\[ 0 = \log_{3}x + 1 \]Solving this gives:\[ x = \frac{1}{3} \]Thus, the x-intercept is at\( \left( \frac{1}{3}, 0 \right) \).The x-intercept is significant because it represents a point where the function transitions between positive and negative outputs relative to the x-axis.
Analyzing intercepts like this one helps in comprehending how a function is positioned within the coordinate plane, aiding in the understanding of its behavior across different ranges.