Problem 78 - Precalculus

Question

Graph the logarithmic function using transformation techniques. State the domain and range of \(f\). $$f(x)=\log _{3}(x+1)-2$$

Step-by-Step Solution

Answer

The domain is \((-1, \infty)\) and the range is \((-\infty, \infty)\).

Step 1: Identify the Parent Function

The parent function for the given function is the basic logarithmic function, \(g(x) = \log_3(x)\). This is the standard form of a logarithmic function without any transformations applied.

Step 2: Apply Transformations

The function \(f(x) = \log_3(x + 1) - 2\) involves two transformations to the parent function \(g(x) = \log_3(x)\). The transformations are:1. Horizontal shift: \(x + 1\) indicates a shift of 1 unit to the left.2. Vertical shift: \(-2\) indicates a downward shift of 2 units.

Step 3: Determine the New Domain

To find the domain, set \(x + 1 > 0\) because the argument of a logarithm must be positive. Solving this inequality, we have \(x > -1\). Thus, the domain of \(f\) is \(x > -1\), or in interval notation, \((-1, \infty)\).

Step 4: Determine the New Range

The range of the parent function \(\log_3(x)\) is \((-\infty, \infty)\). Vertical shifts do not affect the range of a logarithmic function, so the range of \(f(x) = \log_3(x + 1) - 2\) remains \((-\infty, \infty)\).

Step 5: Sketch the Graph

To sketch the graph, start with the basic logarithm \(g(x) = \log_3(x)\). Shift it left by 1 unit (changing the vertical asymptote to \(x = -1\) instead of \(x = 0\)), then move the graph down by 2 units. This gives the transformed function \(f(x)\) with a new asymptote and shifted position.

Key Concepts

Transformations

Logarithmic functions often require transformations to be applied in order to graph them accurately. Transformations help in understanding how these functions shift, stretch, or compress on the graph. For the function \(f(x) = \log_3(x + 1) - 2\), two essential transformations occur:

• Horizontal Shift: The presence of \(x + 1\) in the logarithmic function indicates a horizontal shift. Specifically, it's shifted 1 unit to the left. This transformation alters where the graph starts, impacting the vertical asymptote. For the parent function \(g(x) = \log_3(x)\), the vertical asymptote is at \(x = 0\). After shifting, it moves to \(x = -1\).


• Vertical Shift: The subtraction of 2 from the function signifies a downward vertical shift by 2 units. While this shift influences where the function will "sit" on the graph in terms of its height, it does not affect the vertical asymptote or the horizontal placement.

Understanding these transformations is crucial for accurately graphing the function and imagining its new position on the graph.

Domain and Range

The domain and range of logarithmic functions are vital in determining where the function exists and how it behaves.

• Domain: For any logarithmic function, only positive real numbers are allowed as inputs to the logarithm. This stems from the fact that the logarithm of a negative number or zero is undefined. For our function \(f(x) = \log_3(x + 1) - 2\), we set \(x + 1 > 0\) to find the domain. Solving this inequality, \(x > -1\) gives us the domain. In interval notation, it is expressed as \((-1, \infty)\).


• Range: The range of a logarithmic function like \(g(x) = \log_3(x)\) is all real numbers, \((\-\infty, \infty)\). Despite the vertical shift in \(f(x)\), the growth of a logarithmic function remains infinite in both directions when considering its range. Thus, the range, in this case, remains \((\-\infty, \infty)\).

Mastering the concepts of domain and range helps ensure the mathematical integrity of the function and helps predict its behavior on the graph.

Graph Sketching

Graph sketching of a logarithmic function involves visualizing and plotting the transformations applied to the parent function. Here’s a step-by-step process to effectively sketch the given function:

• Start with the Parent Function: Begin by sketching the basic form of the parent function \(g(x) = \log_3(x)\). The graph typically starts from the vertical asymptote at \(x = 0\) and rises gradually, approaching infinity as \(x\) increases.


• Apply the Horizontal Shift: Shift the entire graph of the parent function 1 unit to the left. This shift is essential to align the graph with the function \(f(x) = \log_3(x + 1) - 2\). The new vertical asymptote will be at \(x = -1\).


• Apply the Vertical Shift: Move the graph down by 2 units. This shift does not alter the overall orientation of the graph but ensures that the vertical alignment corresponds to the function's equation.

Incorporating these steps allows for a clear sketch of the logarithmic function, showcasing its progression and changes at each transformation step. Remember to note the asymptote and ensure smoothness in the curve as it transitions from the original parent function.