what is the equation of the exponential function represented by the graph?$\boldsymbol{f(x)=2(4)^x}$$\boldsym…

Explanation:

Step1: Recall exponential form

The general form of an exponential function is $f(x)=ab^x$, where $a$ is the y-intercept, $b>0, b\neq1$.

Step2: Identify y-intercept from graph

The graph crosses the y-axis at $(0,2)$, so $a=2$.

Step3: Test a second point

Pick a point on the graph, e.g., $(-1,4)$. Substitute $x=-1$, $f(x)=4$, $a=2$ into $f(x)=ab^x$: $4=2b^{-1}$ Solve for $b$: $b^{-1}=\frac{4}{2}=2$ $b=\frac{1}{2}$

Step4: Verify the function

Substitute $a=2$, $b=\frac{1}{2}$ into the general form: $f(x)=2\left(\frac{1}{2}\right)^x$. Check if other points match: when $x=1$, $f(1)=2\left(\frac{1}{2}\right)^1=1$, which aligns with the graph.

Answer:

$\boldsymbol{f(x)=2\left(\frac{1}{2}\right)^x}$ (the third option)