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Find the formulas for exponential functions

g(0)=6 and g(-2)=12
h(.5)=4 and h(.25)=16
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1 Answer

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