Question

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

Accepted Answer

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 $$(x+3). $$Determine the domain of the function by finding the values of $$x$$ for which the logarithm is defined. Since $$\log_{2}(x+3)$$ requires $$x+3 \u003e 0$$, solve the inequality $$x + 3 \u003e 0 to $$get $$x \u003e -3. $$Thus, the domain is $$(-3, \infty). $$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 $$0  1. $$This corresponds to $$-3  -2$$ respectively. 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 $$\log_{2}(x+3). $$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.

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

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Key Concepts

Logarithmic Functions

Absolute Value Function

Domain and Range of Functions