Q. 85 - Precalculus Enhanced with Graphing Utilities

Question

Determine the exponential function whose graph is given.

Step-by-Step Solution

Answer

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

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 $f\left(x\right) = 3^x$.