Final answer:
To find the exponential function f(x) = ae^kx that passes through the points (2,12) and (4,4/3), we set up and solved a system of equations. The solution is the function f(x) = 108e^(ln(1/9)/2)x.
Step-by-step explanation:
To write an exponential function in the form f(x) = aekx that passes through the points (2,12) and (4,4/3), we need to find the constants 'a' and 'k'. We can set up a system of two equations using these points:
- 12 = ae2k (1)
- 4/3 = ae4k (2)
Dividing equation (2) by equation (1), we get:
(4/3) / 12 = (ae4k) / (ae2k)
1/9 = e2k
Taking the natural logarithm of both sides gives us
ln(1/9) = 2k
k = ln(1/9)/2
Now we can substitute the value of 'k' into equation (1) to find 'a':
12 = a * e(ln(1/9)/2)*2
12 = a * eln(1/9)
12 = a / 9
a = 12 * 9
Therefore, the exponential function that passes through the given points is
f(x) = 108e(ln(1/9)/2)x.