Graph each function. ƒ(x) = log₃ (x-1) + 2

Verified step by step guidance

1. Identify the base of the logarithm and the transformations applied to the function. Here, the function is \(f(x) = \log_{3}(x - 1) + 2\), which is a logarithmic function with base 3, shifted horizontally and vertically.
2. Determine the domain of the function by setting the argument of the logarithm greater than zero: \(x - 1 > 0\). Solve this inequality to find the domain.
3. Find the vertical asymptote by setting the argument of the logarithm equal to zero: \(x - 1 = 0\). This vertical line is where the function approaches infinity or negative infinity.
4. Identify the horizontal shift and vertical shift. The \(x - 1\) inside the logarithm shifts the graph 1 unit to the right, and the \(+2\) outside shifts the graph 2 units upward.
5. Plot key points by choosing values of \(x\) greater than 1, calculate corresponding \(f(x)\) values using the function, and sketch the graph showing the vertical asymptote at \(x=1\), the general shape of the logarithmic curve, and the shifts.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as f(x) = log_b(x), where b is the base. It answers the question: to what power must the base b be raised to produce x? Understanding the properties of logarithms is essential for graphing and interpreting these functions.

Domain of Logarithmic Functions

The domain of a logarithmic function f(x) = log_b(x - h) is all x-values for which the argument (x - h) is positive. For example, in f(x) = log_3(x - 1) + 2, the domain is x > 1 because the logarithm is undefined for zero or negative inputs. Identifying the domain helps in correctly plotting the graph.

Transformations of Functions

Transformations shift or change the shape of a graph. For f(x) = log_3(x - 1) + 2, the (x - 1) inside the log shifts the graph 1 unit to the right, and the +2 outside shifts it 2 units up. Recognizing horizontal and vertical shifts is crucial for accurately graphing the function.