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

Verified step by step guidance

1. Step 1: Understand the function f(x) = \log_{1/2}(x) - 2. This is a logarithmic function with base 1/2, shifted down by 2 units.
2. Step 2: Determine the domain of the function. Since the logarithm is only defined for positive values of x, the domain is x > 0.
3. Step 3: Determine the range of the function. Logarithmic functions can take any real number as output, so the range is all real numbers, (-∞, ∞).
4. Step 4: Identify key points for graphing. For example, when x = 1, \log_{1/2}(1) = 0, so f(1) = 0 - 2 = -2. When x = 1/2, \log_{1/2}(1/2) = 1, so f(1/2) = 1 - 2 = -1.
5. Step 5: Sketch the graph. The graph will be a downward-sloping curve that approaches negative infinity as x approaches 0 from the right, and it will pass through the points identified in Step 4.

Key Concepts

Logarithmic Functions

Logarithmic functions, such as f(x) = log_b(x), are the inverses of exponential functions. They are defined for positive real numbers and have a base 'b' that determines their growth rate. In this case, the function f(x) = log_(1/2)(x) indicates a logarithm with a base of 1/2, which means it decreases as x increases, reflecting the properties of logarithms with bases less than 1.

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 (f(x)). For the function f(x) = log_(1/2)(x) - 2, the domain is x > 0, since logarithms are only defined for positive numbers. The range, however, is all real numbers, as logarithmic functions can take on any value as x approaches 0 or increases indefinitely.

Graphing Logarithmic Functions

Graphing logarithmic functions involves plotting points based on the function's values and understanding its general shape. The graph of f(x) = log_(1/2)(x) - 2 will show a decreasing curve that approaches negative infinity as x approaches 0 and shifts downwards by 2 units due to the '-2' in the function. Key features to note include the vertical asymptote at x = 0 and the intercepts, which help in accurately sketching the graph.