Is the set of functions f1 (x) = ex + 2, f2 (x) = ex - 4 linearly dependent or linearly independent on (-[infinity], [infinity]) ? Discuss. Since ex - 4 = ex + 2, we see that ex - 4 a constant multiple of ex + 2 and the set of functions is linearly.

not linearly independent

Step-by-step explanation:

Given:

- A set of function is given as follows:

f1 (x) = e^ (x + 2)

f2 (x) = e^ (x-4)

Find:

Is the set of functions f1 (x) & f2 (x) linearly dependent or linearly independent on (-[infinity], [infinity]) ?

Solution:

- Re-write the two functions in the form as shown below:

f1 (x) = e^ (x) * e^2

f2 (x) = e^ (x) * e^-4

- Divide the two functions f1 (x) / f2 (x):

f1 (x) / f2 (x) = [e^ (x) * e^2] / [e^ (x) * e^-4]

f1 (x) / f2 (x) = e^ (6)

- The function f1 (x) is e^6*f2 (x) a scalar multiple we can say that two functions are not linearly independent.