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
choose the correct graph below

Explanation:

Step1: Analyze the properties of the logarithmic function \(y = \log_{0.5}x\)

The general form of a logarithmic function is \(y=\log_{a}x\). For \(y = \log_{0.5}x\), since \(0\lt a = 0.5\lt1\), the function is a decreasing function.
The domain of \(y=\log_{a}x\) is \(x>0\) (i.e., \((0,\infty)\)) because the logarithm of a non - positive number is not defined in the set of real numbers. The range of \(y = \log_{a}x\) is \((-\infty,\infty)\) (all real numbers).
To find the \(x\) - intercept, set \(y = 0\). Then \(\log_{0.5}x=0\). Using the property \(\log_{a}1 = 0\) for \(a>0,a
eq1\), we get \(x = 1\). So the \(x\) - intercept is \((1,0)\).
The vertical asymptote of \(y=\log_{a}x\) is \(x = 0\) (the \(y\) - axis).
For the end - behavior: As \(x
ightarrow0^{+}\), \(y=\log_{0.5}x
ightarrow\infty\) (since \(a = 0.5\lt1\)). As \(x
ightarrow\infty\), \(y=\log_{0.5}x
ightarrow-\infty\) (because \(0.5^{y}=x\) and when \(x\) gets large, \(y\) is negative and its absolute value gets large).

Step2: Compare with the graph options

Option A: The function in option A is a decreasing function, has a \(x\) - intercept at \((1,0)\), a vertical asymptote \(x = 0\), and the end - behavior (as \(x
ightarrow0^{+},y
ightarrow\infty\) and as \(x
ightarrow\infty,y
ightarrow-\infty\)) is consistent with \(y=\log_{0.5}x\).
Option B: The function in option B is an increasing function (\(a>1\) for an increasing logarithmic function \(y = \log_{a}x\)), so it is not the graph of \(y=\log_{0.5}x\).
Option C: The \(y\) - intercept is not a property of \(y=\log_{a}x\) (since \(x>0\) for the domain), so this option is incorrect.
Option D: The function in option D is an increasing function, so it is not the graph of \(y=\log_{0.5}x\).

Answer:

A.