Problem 67 - 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. $$g(x)=2 \log _{3}(x-1)$$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = 2 \log_3(x - 1)\) is \(x > 1\) and it has a vertical asymptote at \(x = 1\). In the graph, you will notice the function start at (or never reaches) the vertical asymptote and increases with a factor of 2.

Step 1: Find the Domain of the Logarithmic Function

Firstly, it's important to note a key property of logarithms: they are defined only for positive values. Hence, the expression inside the log can only be greater than 0. For the function \(g(x) = 2 \log_3(x - 1)\), this implies the domain to be \(x > 1\).

Step 2: Identify the Asymptote

Asymptotes provide clear borders where the graph of a function never reaches. For the noted function, there is a vertical asymptote at \(x = 1\) which has been deduced from the domain of the function.

Step 3: Plot the Function

The function can be graphed based on its nature, being a logarithmic function. It will always start from the vertical asymptote at \(x = 1\) and grow/decay based on its coefficient, in this case \(2\). Using a graphing tool or manual plotting gives a clear visual representation of this function.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial because it tells us all possible input values for that function. For logarithmic functions such as \( g(x) = 2 \log_3(x - 1) \), we need the expression inside the logarithm to be positive. This means we solve the inequality \( x - 1 > 0 \) to find the domain. Solving gives us \( x > 1 \), meaning that \( x \) must be greater than 1 for the function to be valid.

This restriction arises because logarithms are undefined for zero and negative numbers. Always check the base of the logarithm too. The base should be positive and not equal to one, but in this exercise, the base is 3, which is valid. Thus, the domain is all real numbers greater than 1, symbolically represented as \( (1, \infty) \).

Remember, knowing the domain helps us understand where the function exists on the number line.

Vertical Asymptote

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. They indicate where a function 'blows up' or becomes undefined. For the logarithmic function \( g(x) = 2 \log_3(x - 1) \), we derive the vertical asymptote from the domain condition set by the logarithm argument.

Since \( x-1 \) must be greater than zero, we see that \( x = 1 \) is where the function cannot be evaluated. Thus, it creates a vertical asymptote here. Graphically, this means that as \( x \) gets closer to 1 from the right side, the function's value decreases or increases sharply towards negative or positive infinity, but never actually touches the line \( x = 1 \).

This insight is essential when sketching the function, as it shows a boundary where the graph shoots up or down indefinitely.

Graphing Functions

Graphing logarithmic functions involves several crucial steps, each drawing from different elements like the domain and asymptotes. When graphing \( g(x) = 2 \log_3(x - 1) \), we start by considering the vertical asymptote at \( x = 1 \). This means the graph will approach, but never cross, this line.

Next, we recognize that the coefficient 2 in front of the logarithm affects the steepness or vertical stretch of the graph. It means the function will grow twice as fast compared to if the coefficient was just 1.

For visualization:

• Start your graph slightly right of the asymptote \(x=1\).
• Draw the curve increasing or decreasing in line with the logarithmic pattern.

Typically, using graphing software or plotting key points manually can provide an accurate depiction. Knowing the domain, asymptotes, and coefficients assists in bringing the function's graph to life.