Exponential function question that I can't figure out how to do.

Question

Given f(1) = 4 and f(6) = 7 find f(16). This is in the exponential functions section of my book, but I can't figure it out.

Answer 1

Well this is a slightly weird answer, but it works. You can set up a set of simultaneous equations. An exponential function has the general formula f(x) = a * b^x + c

 

So f(1) = 4 means that a*b +c = 4

    f(6) = 7 means that a*b⁶ + c = 7     subtract the 2 equations to get

 

a*b⁶ - a*b = 3    factor out the greatest common factor

ab(b⁵ -1) = 3

 

Now since 3 is prime, you get 2 possibilities 

ab = 1 and b⁵ - 1 = 3                    or         ab = 3 and b⁵ - 1 = 1       solving you get

b = fifth root of 4                                         b = fifth root of 2

a = 1/fifth root of 4                                       a = 3/fifth root of 2

 

and c = 3                                                    and c = 1

 

f(x) = 1/fifth root of 4 *(fifth root of 4)^x + 3      or      f(x) = 3/fifth root of 2(fifth root of 2)^x + 1

 

Both work....check it yourself to see f(1)=4 and f(6)=7 in both cases. so f(16)=

 

f(16) = 67                                                          or   f(16) = 25

 

Hope this helps....weird answers bur they work.