Graph each function. Give the domain and range. ƒ(x) = | log￬2 (x+3) |

Verified step by step guidance

1. Step 1: Understand the function f(x) = | log_2(x+3) |. This is the absolute value of the logarithmic function log base 2 of (x+3).
2. Step 2: Determine the domain of the function. The expression inside the logarithm, (x+3), must be greater than 0. Solve the inequality x+3 > 0 to find the domain.
3. Step 3: Solve the inequality x+3 > 0, which gives x > -3. Therefore, the domain of the function is all real numbers greater than -3, or in interval notation, (-3, ∞).
4. Step 4: Consider the range of the function. Since the function is the absolute value of a logarithm, the output will always be non-negative. The range is [0, ∞).
5. Step 5: To graph the function, plot the basic shape of log_2(x+3) for x > -3, then reflect any negative values above the x-axis to account for the absolute value, ensuring all y-values are non-negative.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function ƒ(x) = | log₂(x + 3) |, the argument of the logarithm, x + 3, must be greater than zero, leading to the condition x > -3. Thus, the domain is all real numbers greater than -3.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. In the case of ƒ(x) = | log₂(x + 3) |, since the logarithm can take any real number value and the absolute value function transforms all outputs to non-negative values, the range is all non-negative real numbers, or [0, ∞).

Graphing Logarithmic Functions

Graphing logarithmic functions involves understanding their general shape and behavior. The function ƒ(x) = log₂(x + 3) will have a vertical asymptote at x = -3 and will increase without bound as x increases. The absolute value modifies the graph, reflecting any negative values above the x-axis, resulting in a graph that is always non-negative.