Problem 63 - 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. $$f(x)=\log _{4}(x+1)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log _{4}(x+1)\) is \(x > -1\). The function has a vertical asymptote at \(x = -1\). When graphing, start from the asymptote and, as \(x\) increases, the value of \(f(x)\) also increases.

Step 1: Find the Domain

The logarithmic function \(f(x)=\log _{4}(x+1)\) is defined for all x such that \(x+1 > 0\). Therefore, subtract 1 from both sides to find that \(x > -1\). This is the domain of the function.

Step 2: Identify the Asymptote

The vertical asymptote of this function is at \(x = -1\). Therefore, on graphing this function, there will be a vertical asymptote at \(x = -1\). It is the line that the function will approach but never touch or cross.

Step 3: Graph the Function

The base of the logarithm (4) is greater than 1, indicating that the function is increasing. As \(x\) moves away from -1 on the right side, \(f(x)\) will increase slowly. Use points in the domain to sketch the graph. For example, when \(x = 0\), \(f(x) = \log_{4}(1) = 0\). So point (0,0) is on the graph. Also, since our function is a logarithmic function shifted horizontally by -1, the main features of the graph (the asymptote and the general direction of the graph) will resemble those of the parent function \(\log_{4}(x)\), but moved 1 unit to the left.

Key Concepts

Logarithmic functions

Logarithmic functions are a fascinating type of mathematical operation that represent the inverse of exponential functions. In simpler terms, while exponential functions might ask, "What is the result when you raise a given base to a certain power?", logarithmic functions reverse this and ask, "To what power must you raise a given base to yield a certain number?" In our example, the function is defined with a logarithm base of 4, which means we seek the power that results in a specific outcome on that base.

Logarithmic functions are only defined for positive values of the argument, in this case, the expression inside the log, which is \(x+1\). Thus, the crucial part of finding the domain of \(f(x) = \log_{4}(x+1)\) is understanding where that expression is greater than zero. This means the domain for our function is \(x > -1\).

Common bases used include 10 (common logarithmic), \e\ (natural logarithmic), and 2 in binary systems. Our base here is 4, making it a less common but equally valid form of logarithm.

Logarithmic functions can also change depending on the characteristics of their inverses: if an exponential function grows rapidly, its logarithmic counterpart will likely grow more slowly.

Vertical asymptotes

Vertical asymptotes are a key feature of many types of mathematical functions, including logarithmic ones. Simply put, a vertical asymptote represents a kind of invisible barrier on a graph. The function gets close to this line but never actually crosses it.

For \(f(x) = \log_{4}(x+1)\), the vertical asymptote is at \x = -1\. At this point, the function approaches negative infinity as \(x\) gets closer and closer to -1 from the right. It's at this line that the value of the function drops off sharply and essentially becomes undefined.

• A vertical asymptote occurs because the expression inside the logarithm can never be zero or negative. Anything less than \(x = -1\) would make \(x + 1\) zero or negative, rendering the logarithmic expression invalid.
• The location of vertical asymptotes can often be determined by setting the argument of the logarithm equal to zero, as we did when solving for the domain.

Function graphing

Function graphing is a method used to visualize how a mathematical function behaves along a coordinate plane. By plotting points for specific \(x\) values and connecting them, one can see the shape and general trend of the function. For \(f(x) = \log_{4}(x+1)\), graphing can help elucidate where the function's key features lie.

The graph of a logarithm function like this one will generally have the following characteristics:

• It passes through the point \(0, 0\) as illustrated because \(\log_{4}(1) = 0\).
• The graph will increase slowly towards infinity since the base of the log is greater than 1 (in this case, 4), making it an increasing function.
• It will approach the vertical asymptote without crossing it, showing a steep decline in value as \(x\) nears \(x = -1\).
• The graph will resemble a shifted version of the basic \(\log_{4}x\) function, effectively moving one unit to the left along the \(x\)-axis due to the \(+1\) inside the argument.

Graphing aids in cementing your understanding because it offers a tangible representation of the concepts at work, like the domain, and how changes in the functional formula affect its shape.