question 1 (essay worth 2 points)\n(05.06 hc)\nthe table represents a logarithmic function f(x).\nuse the…

Explanation:

Step1: Recall logarithmic function properties

The general form of a logarithmic function is (y = \log_{a}x). For a logarithmic function (y=\log_{a}x), the domain is (x>0) and the range is all real - numbers.

Step2: Analyze the given table for domain

Looking at the (x) - values in the table (\left{\frac{1}{128},\frac{1}{64},\frac{1}{8},1,8,64,128\right}), all (x) - values are positive real numbers. In inequality notation, the domain is (x > 0), in interval notation, it is ((0,\infty)), and in set - builder notation, it is ({x|x\in\mathbb{R},x > 0}).

Step3: Analyze the given table for range

The (y) - values in the table are ({- 7,-6,-1,0,1,2,3}). Since a logarithmic function can take on all real - values, in inequality notation, the range is (y\in\mathbb{R}), in interval notation, it is ((-\infty,\infty)), and in set - builder notation, it is ({y|y\in\mathbb{R}}).

Step4: Graph the function

To graph the function, plot the points ((\frac{1}{128}, - 7),(\frac{1}{64},-6),(\frac{1}{8},-1),(1,0),(8,1),(64,2),(128,3)) on the coordinate plane. The graph of a logarithmic function (y = \log_{a}x) passes through the point ((1,0)) and has a vertical asymptote at (x = 0). As (x) approaches (0) from the right, (y) approaches (-\infty), and as (x) approaches (\infty), (y) approaches (\infty).

Answer:

Domain: Inequality notation: (x>0); Interval notation: ((0,\infty)); Set - builder notation: ({x|x\in\mathbb{R},x > 0}) Range: Inequality notation: (y\in\mathbb{R}); Interval notation: ((-\infty,\infty)); Set - builder notation: ({y|y\in\mathbb{R}})