In Exercises 32–35, the graph of a logarithmic function is given. Select the function for each graph from the following options: f(x) = logx, g(x) = log(-x), h(x) = log(2-x), r(x)= 1+log(2-x)

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1. Identify the key characteristics of each function: domain, range, and transformations.
2. For f(x) = \log(x), the domain is x > 0, and the graph passes through (1,0).
3. For g(x) = \log(-x), the domain is x < 0, and the graph is a reflection of \log(x) across the y-axis.
4. For h(x) = \log(2-x), the domain is x < 2, and the graph is a horizontal shift of \log(-x) to the right by 2 units.
5. For r(x) = 1+\log(2-x), the domain is x < 2, and the graph is a vertical shift of h(x) upwards by 1 unit.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The general form is f(x) = log_b(x), where b is the base. These functions exhibit a characteristic shape, increasing slowly and approaching negative infinity as x approaches zero from the right. Understanding their properties, such as domain and range, is crucial for analyzing their graphs.

Domain and Range of Logarithmic Functions

The domain of a logarithmic function is limited to positive values of x, as the logarithm of zero or a negative number is undefined. For example, f(x) = log(x) has a domain of (0, ∞). The range, however, is all real numbers, meaning logarithmic functions can take any value as x increases. Recognizing these constraints helps in selecting the correct function based on the graph provided.

Transformations of Logarithmic Functions

Transformations involve shifting, reflecting, or stretching the graph of a function. For instance, the function r(x) = 1 + log(2 - x) indicates a vertical shift upwards by 1 unit and a horizontal transformation due to the '2 - x' term. Understanding these transformations is essential for interpreting how the graph of a logarithmic function changes in response to modifications in its equation.