Question

Graph each function. Give the domain and range. ƒ(x) = | log1/2 (x-2) |

Accepted Answer

Identify the function given: $$f(x) = \left| \log_{\frac{1}{2}} (x - 2) \right|. $$This is the absolute value of a logarithmic function with base $$\frac{1}{2}$$ and argument $$(x - 2). $$Determine the domain by finding the values of $$x$$ for which the argument of the logarithm is positive: solve $$x - 2 \u003e 0$$, which gives $$x \u003e 2. So$$, the domain is $$(2, \infty). $$Understand the behavior of the logarithm with base $$\frac{1}{2}. $$Since $$\frac{1}{2} \u003c 1$$, the logarithmic function $$\log_{\frac{1}{2}}(x - 2) is a $$decreasing function. It approaches $$+\infty as$$ $$x$$ approaches 2 from the right, and it approaches $$-\infty as$$ $$x$$ goes to infinity. Apply the absolute value to the logarithmic function. This means all negative values of $$\log_{\frac{1}{2}}(x - 2)$$ become positive, so the graph will reflect any part of the logarithm below the $$x-$$axis upwards. Determine the range by considering the absolute value transformation. Since the logarithm can take all real values from $$-\infty to$$ $$+\infty$$, after applying the absolute value, the range becomes $$[0, \infty). $$The function never outputs negative values.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Logarithmic Functions

Domain and Range of Functions

Absolute Value Transformations