Graph each function. Give the domain and range. ƒ(x) = | log￬1/2 (x-2) |

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The function ƒ(x) = log_b(x) is defined only for x > 0, where b is the base of the logarithm. In this case, the base is 1/2, which means the function will decrease as x increases. Understanding the properties of logarithms is essential for determining the domain and behavior of the function.

Absolute Value Functions

The absolute value function, denoted as |x|, measures the distance of x from zero on the number line, resulting in non-negative outputs. In the context of the given function, the absolute value of the logarithm ensures that the output is always non-negative, regardless of whether the logarithm itself is positive or negative. This affects the range of the function significantly.

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values). For the function ƒ(x) = | log_1/2 (x-2) |, the domain is determined by the condition x-2 > 0, leading to x > 2. The range, due to the absolute value, will be all non-negative real numbers, starting from 0 to positive infinity.