exponential functions\nwhich exponential function is represented by the graph?\n$f(x)=2(2)^x$\n$f(x)=\\frac{1…

Explanation:

Step1: Recall exponential form

The general exponential function is $f(x) = ab^x$, where $a$ is the y-intercept (when $x=0$).

Step2: Identify y-intercept from graph

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

Step3: Use another point to find $b$

Use the point $(1,1)$. Substitute $x=1$, $f(1)=1$, $a=\frac{1}{2}$ into $f(x)=ab^x$: $1 = \frac{1}{2}b^1 \implies b = 2$

Step4: Verify with third point

Use $x=-1$, $f(-1)=0.25$. Substitute into $f(x)=\frac{1}{2}(2)^x$: $f(-1)=\frac{1}{2}(2)^{-1} = \frac{1}{2} \times \frac{1}{2} = 0.25$, which matches the graph.

Answer:

$f(x) = \frac{1}{2}(2)^x$