Graph each function. Give the domain and range. ƒ(x) = (log_1/2 x) - 2

Verified step by step guidance

1. Identify the base of the logarithm function. Here, the function is \(f(x) = \log_{\frac{1}{2}} x - 2\), where the base is \(\frac{1}{2}\), which is between 0 and 1, indicating a decreasing logarithmic function.
2. Determine the domain of the function. Since logarithms are only defined for positive arguments, set the inside of the log greater than zero: \(x > 0\). Thus, the domain is \((0, \infty)\).
3. Analyze the vertical shift caused by the \(-2\) outside the logarithm. This shifts the entire graph down by 2 units, affecting the range but not the domain.
4. Find the range of the function. The logarithmic function \(\log_{\frac{1}{2}} x\) can take any real value from \(-\infty\) to \(\infty\), so after shifting down by 2, the range remains \((-\infty, \infty)\).
5. Sketch the graph by plotting key points such as when \(x=1\) (since \(\log_{b} 1 = 0\) for any base \(b\)), which gives \(f(1) = 0 - 2 = -2\). Also, note the behavior as \(x\) approaches 0 from the right (function goes to \(\infty\)) and as \(x\) approaches infinity (function goes to \(-\infty\)) due to the base being less than 1.

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 get x? Understanding the properties of logarithms, especially with bases between 0 and 1, is essential for graphing and analyzing the function.

Domain and Range of Logarithmic Functions

The domain of a logarithmic function f(x) = log_b(x) consists of all positive real numbers because the logarithm of zero or negative numbers is undefined. The range is all real numbers since logarithmic functions can produce any real output. Shifts and transformations affect these sets and must be considered when determining domain and range.

Transformations of Functions

Transformations include shifts, reflections, stretches, and compressions applied to the parent function. For f(x) = log_(1/2)(x) - 2, subtracting 2 shifts the graph downward by 2 units. Understanding how these changes affect the graph helps in accurately plotting the function and identifying its domain and range.