Question

Determine the exponential function whose graph is given.

Short answer

The exponential function is$f\left(x\right)=3^x$.

Step-by-step solution

Step 1. Given Information

Points given on graph:

$(-1,\frac{1}{3}),(1,3),(0,1),(2,9)$

We have to find the exponential of the form $f\left(x\right)=a^x$

where $\frac{f(x+1)}{f\left(x\right)}=a,a>0$

and x is any real number.

Step 2. Calculating the ratios

Put $x=0$

$\frac{f(x+1)}{f\left(x\right)}=\frac{f\left(1\right)}{f\left(0\right)}$

From graph , we get , $f\left(1\right)=3,f\left(0\right)=1$

$\frac{f\left(1\right)}{f\left(0\right)}=\frac{3}{1}=3$

Put x=1 ,

$\frac{f(x+1)}{f\left(x\right)}=\frac{f\left(2\right)}{f\left(1\right)}$

From graph , we get , $f\left(2\right)=9,f\left(1\right)=3$

$\frac{f\left(2\right)}{f\left(1\right)}=\frac{9}{3}=3$

Similarly , we get , ratios for $x=-1,x=2$

The ratios for them is also 3.

Step 3. Getting the final exponential function 

Since the ratios calculated above is 3.

Therefore , $a=3$

Hence , the exponential function is$f\left(x\right)=3^x$.