How do you find the exponential function \[f\left( x \right) = C{a^x}\] whose graph the given points \[\left( {0,2} \right)\] and \[\left( {3,54} \right)\]?

Hint: In the given question, we have been given a function. This function’s graph has been given with two points. This function is an exponential function. We have to find the value of the exponential function. This problem is quite easy if approached correctly. To solve it, we are first going to find the value of the coefficient. Then we are going to plug in the given value of the points so that we can get a relation for the function given.

Complete step-by-step answer:
First, we put in the first value given into the function,
\[f\left( 0 \right) = C{a^0} = 2\]
Hence, \[C = 2\]
Now, we have the function \[f\left( x \right) = 2{a^x}\].
Now we put in the second given value,
\[f\left( 3 \right) = 2 \times {a^3} = 54\]
Thus, \[{a^3} = 27 \Rightarrow a = 3\]
Hence, we have the exponential function \[f\left( x \right) = 2\left( {{3^x}} \right)\]

Note: In the given question, we had to calculate the value of a given exponential function. We solved it by first finding the value of the coefficient by eliminating the term containing the variable. Then we plugged in the given value of the points so that we get a relation for the function given. So, it is really important that we know the formulae and where, when and how to use them so that we can get the correct result.