Find the formulas for exponential functions g06 and g212 and

Find the formulas for exponential functions
g(0)=6 and g(-2)=12
and
h(.5)=4 and h(.25)=16

Solution

For the first problem:
The ratio g(0)/g(-2) = 1/2
This suggests that we look for g of the form g(x) = A 2-x/b for some value of A and b.
We know there should be a minus sign in the exponent because g(0) is less than g(-2). Of course A cancels out in the ratio g(0)/g(-2). Thus that ratio is just 20/22/b which must evaluate to 1/2. By inspection we see that this implies that b = 2. So g(x) = A 2-x/2 for some value of A. However,
g(0) = A 2-0/2 = 6. So A = 6. Finally

g(x) = 6 2-x/2

For the second problem:
h(.5) / h(.25) = 1/4 = 1/22
As above this suggests that we look for h of the form h(x) = B 2-x/d for some value of B and d.
Proceeding as above h(.5)/h(.25) = 2-.5/d/2-.25/d = 2-.25/d = 1/22 , by inspection d = .125
So h(x) = B 2-8 x and h(.5) = B 2-4 which must equal 4. Thus B = 64.
Finally h(x) = 64 2-8 x
Along the way I used the properties of exponential forms and 1/.125 = 8