Find a formula of the form f( x ) = a ⋅ b^x for the exponential function which satisfies the given conditions: f(3)=1/81 and f(−1)=81.

Question

I can't seem to get it

Answer 1

f(3) = 1/81 ---> a * b^3 = 1/81

f(-1) = a/b = 81 ---> b/a = 1/81

so a*b^3 = b/a

a^2b^2 = 1

(ab)^2 - 1 = 0

(ab+1)(ab-1) = 0

ab = -1 or ab=1

which means a is either the reciprocal of b OR a is the NEGATIVE reciprocal of b...

if they are just reciprocals, then

f(x) = k (1/k)^x

f(3) = k(1/k)^3 = 1/81

1/k^2 = 1/81

k^2 = 81

k = +or- 9

f(x) = 9(1/9)^x

or

f(x) = -9( -1/9)^x