Graph each function. Give the domain and range. ƒ(x) = | log₂ (x+3) |

Verified step by step guidance

1. Identify the function given: \(f(x) = \left| \log_{2}(x+3) \right|\). This is the absolute value of a logarithmic function with base 2 and argument \((x+3)\).
2. Determine the domain of the function by finding the values of \(x\) for which the logarithm is defined. Since \(\log_{2}(x+3)\) requires \(x+3 > 0\), solve the inequality \(x + 3 > 0\) to get \(x > -3\). Thus, the domain is \((-3, \infty)\).
3. Analyze the behavior of the logarithmic function \(\log_{2}(x+3)\) before applying the absolute value. Recall that \(\log_{2}(x+3)\) is negative when \$0 < x+3 < 1\(, zero when \)x+3 = 1\(, and positive when \)x+3 > 1\(. This corresponds to \)-3 < x < -2\(, \)x = -2\(, and \)x > -2$ respectively.
4. Apply the absolute value to the logarithmic function. For \(x\) values where \(\log_{2}(x+3)\) is negative, the function \(f(x)\) will reflect those values above the x-axis, making them positive. For \(x\) values where \(\log_{2}(x+3)\) is zero or positive, \(f(x)\) remains the same as \(\log_{2}(x+3)\).
5. Determine the range of \(f(x)\). Since the absolute value makes all outputs non-negative and the logarithmic function can grow without bound positively, the range is \([0, \infty)\). To graph, plot key points such as \(x = -3\) (approaching domain boundary), \(x = -2\) (where \(f(x) = 0\)), and values greater than \(-2\) to see the increasing behavior.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as log_b(x), where b is the base. It is defined only for positive arguments (x > 0). Understanding how to evaluate and graph log functions, including their domain restrictions, is essential for analyzing ƒ(x) = |log₂(x+3)|.

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x, reflecting any negative inputs to positive values. When applied to a function, it affects the graph by flipping all negative outputs above the x-axis, which impacts the range and shape of ƒ(x) = |log₂(x+3)|.

Domain and Range of Functions

The domain is the set of all input values for which the function is defined, while the range is the set of possible output values. For ƒ(x) = |log₂(x+3)|, determining the domain involves solving inequalities inside the logarithm, and the range is influenced by the absolute value transformation.