5. Find the equation of the function f(x) passing through the points (3,8) and (10,22) if f(x) is exponential?

Answer 1

Assuming you simply meant y=a*b^x, I ended up with y=5.1856660671*1.15547835217^x

This is found by solving for the two equations from your question:
a*b^3 = 8
a*b^10 = 22

Yes, I cheated and used my calculator, which says that those long decimal numbers in the answer above are actually equal to:
a = (8/11) * 2^(6/7) * 11^(4/7)
b = (1/2) * 2^(5/7) * 11^(1/7)

Answer 2

By F(x) being exponential I assume you mean f(x) = a*exp(bx).

Take the logarithm Ln[f(x)] = Ln(a) + bx Make two equations by plugging in the points and solving the system of equations.

Ln8 =Ln(a) +3b
Ln22 = Ln(a) +10b

Answer 3

What exactly is the function?
Assume y=exp(x)
ln (y)= x
This is of form yprime=mx+b where you use ln yprime instead of y .
with x1=3 y1=ln(8); x2=10 y2=ln(22)
slope=(y2-y1)/(x2-x1) = m
yprime=mx+b
you have points 3, ln(8) and 10, ln(22)
ln(22)=3m+b
solve for b as you know m.
then the equation yprime=mx+b is known.

Answer 4

f(x) = a*e^bx
8= a*e^3b
22=a e^10b
dividing
11/4 = e^7b so taking ln
b=1/7*ln (11/4)= 0.1445
a=8/e^3b=5.1857
The function f(x) = 5.1857 * e^0.1445x