Find the formulas for exponential functions

Question

g(0)=6 and g(-2)=12

 

 

and

 

h(.5)=4 and h(.25)=16

Answer 1

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 2⁰/2^2/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