Problem 72 - 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. $$g(x)=\ln \left(x^{2}\right)$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \ln(x^2)\) is \(x != 0\) and it carries a vertical asymptote at \(x = 0\).

Step 1: Find the Domain of the Function

The logarithmic function \(g(x) = \ln(x^2)\) is only defined for positive numbers. Therefore, we set the argument of the logarithm namely \(x^2 > 0\). This means the domain of this function is \(x < 0\) or \(x > 0\) or put together as \(x != 0\). The function is undefined at \(x = 0\).

Step 2: Plot the Function

We can now plot \(g(x) = \ln(x^2)\). It's helpful to plot a few key points to see the shape of the graph. We'll see that for x less than 0, the plot is a reflection of the plot for x greater than 0.

Step 3: Identify the Asymptotes

Now, we identify the vertical asymptotes of the function. These are the lines that the function approaches but never crosses. From the domain of the function, we identified that the function is undefined at \(x = 0\), so there exists a vertical asymptote there. Thus the asymptote is \(x = 0\).

Key Concepts

Domain of a Function

Understanding the domain of a function is fundamental in mathematics, particularly when dealing with logarithmic functions. Put simply, the domain is the set of all possible input values (usually x values) for which the function is defined. In the case of logarithmic functions, such as \(g(x) = \backslash ln(x^2)\), the domain is restricted because logarithms are only defined for positive numbers.

We ascertain the domain by setting the inside of the logarithm greater than zero, because you cannot take the logarithm of zero or a negative number without venturing into complex numbers, which are beyond the scope of most logarithmic function discussions. Here, \(x^2 > 0\) leads us to the conclusion that the domain excludes zero (\(x eq 0\)) and includes all other real numbers. This concept is vital as it informs us about the 'legal moves' we can make with the function, much like knowing which squares a chess piece can move to on a chessboard.

Graphing Logarithmic Functions

Graphing logarithmic functions can initially seem daunting, but understanding the process and the properties of logarithms can make it much easier. For our example, \(g(x) = \backslash ln(x^2)\), you start by realizing that the logarithm 'flattens out' as x gets larger, because the rate of increase of the logarithm decreases. It's not a linear relationship.

To graph \(g(x)\), begin with plotting a few key points and observing the symmetry, since \(x^2\) is the same for positive and negative values of x. However, keep in mind the logarithmic function is not defined for zero. An easy way to plot is to select positive x values, compute the corresponding y values, and then reflect these across the y-axis (excluding the line \(x=0\)). Remember, the graph will never touch the x-axis, as the logarithm approaches negative infinity as its argument approaches zero.

Vertical Asymptotes

Vertical asymptotes are like the 'boundaries' of a function's graph. They are lines that the graph approaches infinitely close to but will never actually reach or cross. In the context of logarithmic functions, vertical asymptotes occur where the function is undefined.

For \(g(x) = \backslash ln(x^2)\), the vertical asymptote is at \(x = 0\), because the logarithm isn't defined for zero. Imagine approaching zero on the graph: the function decreases without bound, getting closer and closer to this invisible line at \(x = 0\) without touching it. Identifying vertical asymptotes is crucial, as they are key characteristics of the function's graph and help us understand the behavior of the function at its limits.