Question

which function is represented by the graph below?
$f(x) = \log_{0.5}x$
$f(x) = e \cdot 0.5^x$
$f(x) = e^{0.5x}$
$f(x) = \ln_{0.5}x$

Explanation:

Step1: Analyze the general form of logarithmic function

The general form of a logarithmic function is \(y = \log_{a}x\). When \(0\lt a\lt1\), the function \(y=\log_{a}x\) is a decreasing function. The domain of \(y = \log_{a}x\) is \(x>0\), and the range is all real numbers.

Step2: Analyze the general form of exponential function

The general form of an exponential function is \(y = a^{x}\). When \(0\lt a\lt1\), \(y=a^{x}\) is a decreasing function with domain \(\mathbb{R}\) and range \(y>0\). The function \(y = e^{-0.5x}=(e^{- 0.5})^{x}\approx0.6065^{x}\) is an exponential function. The function \(y = e^{0.5x}=(e^{0.5})^{x}\approx1.6487^{x}\) is an increasing exponential function.

Step3: Analyze the general form of linear - like function

The function \(y = e\cdot0.5x\) is a linear function of the form \(y = kx\) (\(k = 0.5e\approx1.359\)), which is a straight - line passing through the origin.

Step4: Analyze the domain and behavior of \(y=\log_{0.5}x\)

For the function \(y = \log_{0.5}x\), when \(x = 1\), \(y=\log_{0.5}1 = 0\). As \(x\) approaches \(0\) from the right (\(x
ightarrow0^{+}\)), \(y=\log_{0.5}x
ightarrow+\infty\). As \(x\) approaches \(+\infty\), \(y=\log_{0.5}x
ightarrow-\infty\), which matches the shape of the given graph (a decreasing function with domain \(x>0\)).

Answer:

\(f(x)=\log_{0.5}x\)