Final answer:
The values of a and b for the exponential function f(x)=ab^x that passes through (0,7) and (2,112) are a=7 and b=4, found by substituting the points into the function and solving the resulting equations.
Step-by-step explanation:
A student asked about finding the values of a and b for the exponential function f(x)=abx given that it passes through the points (0,7) and (2,112). The process to find the values of a and b involves setting up two equations based on the points given and solving the system of equations.
For the point (0,7), if we substitute x=0 and f(x)=7 into the function, we get:
7 = a*b0 = a*1
Thus, we deduce that a = 7.
Next, using the point (2,112) and the value of a just found, we substitute x=2 and f(x)=112 into the function:
112 = 7*b2
To find b, we divide both sides by 7:
b2 = 112/7
b2 = 16
And taking the square root of both sides gives b = 4, since we typically consider only the positive root for b in this context.
Therefore, the exponential function with the base b is f(x) = 7*4x.