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

and

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

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

and

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

Tutors, please sign in to answer this question.

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 2^{0}/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 = 1/2^{2}

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} = 1/2^{2} , 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

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