Question

How do I find the equation of an exponential function that passes through the point (4,2)?

Answer 1

First of all, we will write a general exponential function which is given by $y = {a^x}$ and then just put in $x = 4$ and $y = 2$ in it to get the required answer.

Complete step by step answer:
We are given that we are required to find an exponential function that passes through the point (4, 2).
Since, we know that a general exponential function is given by $y = {a^x}$. ………..(1)
Now, since we are already given that the function passes through (4, 2). Therefore, we will now just put in $x = 4$ and $y = 2$ in this equation.
Putting $x = 4$ and $y = 2$ in the equation given by $y = {a^x}$, we will then obtain the following equation:-
$\Rightarrow 2 = {a^4}$
Taking the power of $\dfrac{1}{4}$ on the both sides of the above equation, we will then obtain the following equation wits us:-
$\Rightarrow {\left( {{a^4}} \right)^{\dfrac{1}{4}}} = {2^{\dfrac{1}{4}}}$
Simplifying the above equation further, we will then obtain the following equation with us:-
$\Rightarrow a = {2^{\dfrac{1}{4}}}$
Putting this value of a in equation number 1, we will then obtain the following equation:-
$\Rightarrow y = {\left( {{2^{\dfrac{1}{4}}}} \right)^x}$
Simplifying the above equation further, we will then obtain the following equation with us:-
$\Rightarrow y = {2^{\dfrac{x}{4}}}$
Thus, we have the required answer.

Note: The students must note that we have used the inlying fact which is given by the expression given as follows:-
$\Rightarrow {\left( {{a^b}} \right)^c} = {a^{bc}}$
Therefore, when we solved ${\left( {{a^4}} \right)^{\dfrac{1}{4}}}$, we obtained a and when we simplified the term ${\left( {{2^{\dfrac{1}{4}}}} \right)^x}$, we obtained ${2^{\dfrac{x}{4}}}$.
The students must remember that a general exponential function is given by the following equation: $y = {a^x}$.