A function f is described by f (x) = A*exp (kx) + B, where A, B and k are constants. Given f (0) = 1, f (1) = 2, and that the horizontal asymptote of f is - 4, the value of k is

k = ln (6/5)

Step-by-step explanation:

for

f (x) = A*exp (kx) + B

since f (0) = 1, f (1) = 2

f (0) = A*exp (k*0) + B = A+B = 1

f (1) = A*exp (k*1) + B = A*e^k + B = 2

assuming k>0, the horizontal asymptote H of f (x) is

H = limit f (x), when x→ (-∞)

when x→ (-∞), limit f (x) = limit (A*exp (kx) + B) = A * limit [exp (kx) ]+B * limit = A*0 + B = B

since

H = B = (-4)

then

A+B = 1 → A=1-B = 1 - (-4) = 5

then

A*e^k + B = 2

5*e^k + (-4) = 2

k = ln (6/5),

then our assumption is right and k = ln (6/5)