Question

How do you find the formula for the exponential function in the form of $f(x) = C{a^x}$ given $f\left( 0 \right) = 3$ and $f\left( 1 \right) = 15$?

Answer 1

Hint: In this question, we have an exponential function; we have to evaluate the given input. Here, the given information is about the exponential function without knowing the function explicitly. We must use the information to first write the form of the function and then determine the constants. Here, the constants are C and a and evaluate the function.

Complete step by step solution:
In this question, we have an exponential function and we did not know the original function.
The exponential function is:
$ \Rightarrow f(x) = C{a^x}$ ...(1)
Now, the other thing is given that $f\left( 0 \right) = 3$.
Substitute the value of x is equal to 0 in the equation (1).
$ \Rightarrow f(0) = C{a^0}$
But we know that the value of $f\left( 0 \right)$ is 3.
Let us put this value in the above expression.
$ \Rightarrow 3 = C{a^0}$
Now, we know that if the power of any base is 0 then the answer will be 1. The formula is ${x^0} = 1$.
Therefore,
$ \Rightarrow 3 = C\left( 1 \right)$
That is equal to,
$ \Rightarrow C = 3$
So, the value of C is 3.
Now, in question, given that $f\left( 1 \right) = 15$.
Substitute the value of x is equal to 1 in the equation (1).
$ \Rightarrow f(1) = C{a^1}$
But we know that the value of $f\left( 1 \right)$ is 15.
Let us put this value in the above expression.
$ \Rightarrow 15 = C{a^1}$
Let us put the value of C is 3 in the above expression.
$ \Rightarrow 15 = 3 \times {a^1}$
Put ${a^1}$ is equal to a.
$ \Rightarrow 15 = 3 \times a$
Let us divide both sides by 3.
$ \Rightarrow \dfrac{{15}}{3} = \dfrac{{3 \times a}}{3}$
That is equal to,
$ \Rightarrow a = 5$
So, the value of a is equal to 5.
Put the value of a and C in the equation (1).
$ \Rightarrow f(x) = 3 \times {5^x}$

Hence, the formula for the exponential function is $f(x) = 3 \times {5^x}$.

Note: There is another method to solve this question.
Given that $f\left( 0 \right) = 3$.
So, put (0,3) in the equation.
$ \Rightarrow f(x) = C{a^x}$
$ \Rightarrow 3 = C{a^0}$
Now, we know that if the power of any base is 0 then the answer will be 1. The formula is ${x^0} = 1$.
Therefore,
$ \Rightarrow 3 = C\left( 1 \right)$
That is equal to,
$ \Rightarrow C = 3$
So, the value of C is 3.
Given that $f\left( 1 \right) = 15$.
So, put (1,15) in the equation.
$ \Rightarrow f(x) = C{a^x}$
$ \Rightarrow 15 = C{a^1}$
Let us put the value of C is 3 in the above expression.
$ \Rightarrow 15 = 3 \times {a^1}$
Put ${a^1}$ is equal to a.
$ \Rightarrow 15 = 3 \times a$
Let us divide both sides by 3.
$ \Rightarrow \dfrac{{15}}{3} = \dfrac{{3 \times a}}{3}$
That is equal to,
$ \Rightarrow a = 5$
So, the value of a is equal to 5.
Put the value of a and C in the equation (1).
$ \Rightarrow f(x) = 3 \times {5^x}$