Question

How do you find formulas for the exponential functions satisfying the given conditions g(1/2)=4 and g(1/4)=2sqrt2?

Answer 1

An exponential function is generally of the form `y = ab^x`. So, knowing the inputs/outputs of the function, we can write a system of equations with respect to `a` and `b` .

`4 = ab^(1/2)`
`2sqrt(2) = ab^(1/4)`

Solve for `a` in equation `1`.

`a = 4/(b^(1/2))`

`2sqrt(2) = 4/(b^(1/2))b^(1/4)`

`2sqrt(2) = 4/b^(1/4)`

`b^(1/4) = 4/(2sqrt(2))`

`b^(1/4) = 2/sqrt(2)`

`b = (2/sqrt(2))^4`

`b= 16/4`

`b = 4`

Resubstitute to solve for `a`.

`4 = a(4)^(1/2)`

`4 = a(2)`

`a = 2`

Hence, the equation is `y = 2(4)^x`.

Hopefully this helps!