Final answer:
The student's question pertains to determining linear independence of functions in Calculus, specifically within the context of differential equations. Linear independence involves functions not being representable as linear combinations of each other, which can be checked via various tests such as the Wronskian determinant within the given interval.
Step-by-step explanation:
To determine whether a set of functions is linearly independent on the interval (-∞, ∞), we need to use concepts from Calculus. Linear independence of a set of functions is a concept that is often tested in differential equations; it involves ensuring that no function in the set can be written as a linear combination of the other functions in the set. In simpler terms, this means that we require each function to contribute something unique that can't be replicated by a mixture of the others.
For instance, when checking if the functions f(x) = x2 and g(x) = x are linearly independent, we look at the Wronskian determinant or use another test that involves their derivatives. If this determinant is non-zero for all x in the interval, the functions are linearly independent. Otherwise, if the determinant is zero, the functions may be dependent. It is important to note that for functions to be linearly independent, this criterion must be satisfied for all x within the given interval.
Linear Equations such as y = mx + b like those presented in Practice Test 4 's options A, B, and C are also simple examples of linear entities that are independent provided that they are not multiples of each other.