Question

which exponential function is represented by the graph?
$f(x) = 2\left(\frac{1}{2}\
ight)^x$
$f(x) = \frac{1}{2}(2)^x$
$f(x) = \frac{1}{2}\left(\frac{1}{2}\
ight)^x$
$f(x) = 2(2)^x$

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b)^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (determines growth/decay).

Step2: Find \( a \) using \( x = 0 \)

From the graph, when \( x = 0 \), \( y = 0.5 \). Substitute \( x = 0 \) into \( f(x) = a(b)^x \): \( f(0)=a(b)^0=a(1)=a \). So \( a = 0.5=\frac{1}{2} \).

Step3: Find \( b \) using another point

Take the point \( (1, 1) \). Substitute \( x = 1 \), \( y = 1 \), and \( a=\frac{1}{2} \) into \( f(x)=\frac{1}{2}(b)^x \): \( 1=\frac{1}{2}(b)^1 \). Solve for \( b \): \( b = 1\div\frac{1}{2}=2 \).

Step4: Identify the function

Using \( a=\frac{1}{2} \) and \( b = 2 \), the function is \( f(x)=\frac{1}{2}(2)^x \).

Answer:

\( f(x)=\frac{1}{2}(2)^x \) (the second option: \( f(x)=\frac{1}{2}(2)^x \))