Find a possible formula for f(x) if f(1)=6 and f(2)=30 and: (a.) f(x) is a linear function (b.) f(x) is a power function (c.) f(x) is an exponential function

Question

Find a possible formula for f(x) if f(1)=6 and f(2)=30 and:

(a.) f(x) is a linear function

(b.) f(x) is a power function

(c.) f(x) is an exponential function

Answer 1

a) y = mx+b

6 = m+b

30 = 2m+b

subtracting the 1st from the 2nd equation

24 = m

6 = 24+b

b = -18

f(x) = 24x-18

b) y = ax²+b

6 = a+b

30 = 4a+b

24 = 3a

a = 8

6=8 + b

b = -2

f(x) = 8x²-2

c) y = ae^bx

6 = ae^b

30 = ae^2b

ln(6) = ln(a) + b

ln(30) = ln(a) + 2b

b = ln(30) - ln(6) = ln(5)

ln(a) = ln(6) - ln(5)

ln(a) = ln(6/5)

a = 6/5

y = (6/5)e^ln(5)x

= (6/5)5^x

f(x) = 6(5)^x-1