Question

Find a formula for the exponential function which satisfies the given conditions:
f(3)=1/4 and f(−1)=4.

Answer 1

Exponential function:

f(x) =a bˣ


f(3) = a b³ = 1 / 4

f(-1) = a b⁻¹ = 4


a b³ = 1 / 4
÷
a b⁻¹ = 4
__________

b⁴ = 1 / 16

b = ∜1/16

b = 1 / 2

Substitute that into:

a b³ = 1 / 4

a ( 1 / 2 )³ = 1 / 4

a ∙ ( 1 / 8 ) = 1 / 4

Divide both sides by 1 / 8

a = ( 1 / 4 ) / ( 1 / 8 )

a = 2

Your exponential function:

f(x) = a bˣ

f(x) = 2 ∙ ( 1 / 2 )ˣ


Proof:

x = 3

f(3) = 2 ∙ ( 1 / 2 )³ = 2 ∙ ( 1 / 8 ) = 1 / 4


x = - 1

f(-1) = 2 ∙ ( 1 / 2 ) ⁻¹ = 2 / ( 1 / 2 ) = 4