Problem 61 - 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. $$g(x)=\log x-3$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\log x -3\) is \(x>0\). The graph of the function has a vertical asymptote at x = 0, and the function value decreases when x increases. The function approaches but never crosses the vertical line x=0.

Step 1: Find the Domain

Considering \(g(x)=\log x-3\), the argument of the logarithm function (x) must be greater than 0, because you cannot take a log of a negative number or zero. Therefore, the domain of the function is \(x>0\).

Step 2: Identify Asymptotes

Logarithmic functions generally have a vertical asymptote at x = 0. The graph of this function will approach this line as x gets closer to zero, but it will never actually touch or cross it.

Step 3: Graph the Function

To sketch the graph of the function, first, you'll draw the vertical asymptote at x = 0. Next, plot a few points to help shape the graph. For example, when \(x = 1\), \(g(x) = \log(1) - 3 = -3\), and when \(x = 10\), \(g(x) = \log(10) - 3 = 1 - 3 = -2\). This gives an idea of the overall shape of the graph which roughly follows the line of y = x – 3 but never crosses the x = 0 line. For larger x values, the function approaches the line of y = x - 3 without touching it.

Key Concepts

Domain of a Function

The domain of a function is a set of all possible input values that allow the function to have meaningful outputs. For logarithmic functions, the inputs are generally restricted to positive numbers. This is because the logarithm of zero or a negative number is undefined. For our specific function, \(g(x) = \log x - 3\), the input to the logarithm, \(x\), must be greater than zero.
Therefore, the domain of this function is \(x > 0\). You could visualize this as everything to the right of the y-axis on a graph. Ensuring the correct domain is crucial, as it sets the boundary for all other aspects, such as graphing and identifying asymptotes. This fundamental concept helps to ensure that our calculations and representations are accurate.

Vertical Asymptote

A vertical asymptote is a line that a graph of a function approaches but never actually touches or crosses. It is like an invisible barrier guiding the shape of the graph. Vertical asymptotes commonly appear with logarithmic functions due to their inherent properties.
In our context with \(g(x) = \log x - 3\), there is a vertical asymptote at \(x = 0\). This happens because as \(x\) approaches zero from the positive side, the value of \(\log x\) decreases without bound.

• As \(x\) gets very close to zero, \(g(x)\) becomes negatively infinite.
• This asymptotic behavior indicates that the graph will steeply decline downward as the value of \(x\) approaches the limit defined by the domain.

This concept is critical when graphing, as it defines areas the graph cannot occupy, thereby influencing its shape.

Graphing Functions

Graphing functions involves plotting points that satisfy the equation and understanding the behavior of the function at specific intervals. For the function \(g(x) = \log x - 3\), begin by noting the vertical asymptote at \(x = 0\) and that the domain only includes \(x > 0\).
To make the graph, you might want to pick some convenient values of \(x\) within the domain to find points that lie on the curve.

• At \(x = 1\), \(g(x) = \log(1) - 3 = -3\).
• At \(x = 10\), \(g(x) = \log(10) - 3 = 1 - 3 = -2\).

These calculations help to plot the curve by providing reference points. The graph will show a gentle slope upwards to the right, always keeping above the line \(y = x - 3\) while never crossing the asymptote at \(x = 0\). By connecting these points smoothly, it creates a picture of the function’s behavior in its designated domain.