this is the graph of an exponential function.\nwrite an equation for the function in the form $f(x) =…

Explanation:

Step1: Identify the y - intercept

The general form of an exponential function is (f(x)=a(b)^{x}). The y - intercept of a function occurs when (x = 0). From the graph, when (x=0), (y = 4). Substitute (x = 0) and (y=f(0)=4) into the function: (f(0)=a(b)^{0}). Since any non - zero number to the power of 0 is 1 ((b^{0}=1) for (b\neq0)), we have (4=a\times1), so (a = 4).

Step2: Find the value of (b)

Now we know that (a = 4), so the function is (f(x)=4(b)^{x}). We need to find another point on the graph to determine (b). Let's look at the point ((1,2)) (from the graph, when (x = 1), (y=2)). Substitute (x = 1), (y = 2) and (a = 4) into the function: (2=4(b)^{1}). This simplifies to (2 = 4b). To solve for (b), divide both sides of the equation by 4: (b=\frac{2}{4}=\frac{1}{2}=0.5).

Step3: Write the equation

Now that we have (a = 4) and (b=\frac{1}{2}), the equation of the exponential function is (f(x)=4\left(\frac{1}{2}\right)^{x}) or (f(x)=4(0.5)^{x}).

Answer:

(f(x) = 4\left(\frac{1}{2}\right)^{x}) (or (f(x)=4(0.5)^{x}))