Graph each function. ƒ(x) = log￬3 (x-1) + 2

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The function ƒ(x) = log₃(x-1) + 2 represents a logarithm with base 3, which means it answers the question: 'To what power must 3 be raised to obtain (x-1)?' Understanding the properties of logarithms, such as their domain and range, is essential for graphing them accurately.

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the given function, the term (x-1) indicates a horizontal shift to the right by 1 unit, while the +2 indicates a vertical shift upward by 2 units. Recognizing these transformations helps in accurately plotting the graph of the function based on its parent function.

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function ƒ(x) = log₃(x-1) + 2, the domain is x > 1, since the logarithm is only defined for positive arguments. The range, however, is all real numbers, as logarithmic functions can take on any value. Understanding these concepts is crucial for graphing the function correctly.