Question

graph the following function, and identify the domain and range. list any intercepts or asymptotes. describe the end behavior.

$y = \log_{0.5}x$

a. $\\{ y|y>0\\}$
b. $\\{ y|-10c. $\\{ y|y<0\\}$
d. the range is the set of all real numbers.

where is/are the asymptote(s)?

a. there is a vertical asymptote at $x = 0$ and a horizontal asymptote at $y=-3$.
b. there is a horizontal asymptote at $y = 0$.
c. there is a vertical asymptote at $x = 0$.
d. there are no asymptotes.

Explanation:

Step1: Analyze the range of logarithmic function

For the general form of a logarithmic function \(y = \log_{a}x\) (\(a>0,a
eq1\)), the range is the set of all real numbers. The function \(y=\log_{0.5}x\) is a logarithmic function. So its range is \(\mathbb{R}\), which means the range is the set of all real numbers.

Step2: Analyze the asymptote of logarithmic function

For the function \(y = \log_{a}x\) (\(a>0,a
eq1\)), the domain is \(x>0\). As \(x\to0^{+}\), \(y=\log_{a}x\to-\infty\) (when \(0 < a<1\)) or \(y=\log_{a}x\to+\infty\) (when \(a > 1\)). The vertical line \(x = 0\) is a vertical asymptote. There is no horizontal asymptote for the function \(y=\log_{a}x\) since \(\lim_{x\to+\infty}\log_{0.5}x=-\infty\) and \(\lim_{x\to0^{+}}\log_{0.5}x=+\infty\)

Answer:

For the range question: D. The range is the set of all real numbers.
For the asymptote question: C. There is a vertical asymptote at \(x = 0\).