Determine if the following pair of functions is linearly independent or dependent on (-infinity sign, infinity sign): f(t) = e^(3x) and g(t)=xe^(3x)

Answer 1

"(-infinity sign, infinity sign)"
means the interval from -∞ to +∞.

If you have f(t) and g(t) and need to know if they are linearly independent, you can calculate the Wronskian, given by the determinant:
W=
| f(t) g(t) |
|f'(t) g'(t)|

If the Wronskian is non-zero for any value of t, then the two functions are independent.
In this case, the Wronskian turns out to be e^(6t) which is non-zero for all values of t.