Final answer:
The given set of functions {f₁(x), f₂(x), f₃(x)} is linearly independent on the interval (-∞,∞).
Step-by-step explanation:
To determine whether the given set of functions is linearly independent on the interval (-∞,∞), we need to check if there are any non-trivial linear combinations of the functions that result in the zero function. In other words, we need to find constants a, b, and c such that af₁(x) + bf₂(x) + cf₃(x) = 0 for all x. In this case, we have f₁(x) = x, f₂(x) = x², and f₃(x) = 5x - 7x². We can set up the following equation: ax + bx² + c(5x - 7x²) = 0.
Expanding and grouping like terms, we have (c - 7c)x² + (a + 5c)x + bx² = 0. Since this equation must hold for all values of x, the coefficients of each term must be equal to zero. This leads to the following system of equations:
a + 5c = 0
b = 0
c - 7c = 0
Simplifying these equations, we find that a = 0, b = 0, and c = 0. Thus, the only solution to the system is the trivial solution, and the set of functions {f₁(x), f₂(x), f₃(x)} is linearly independent on the interval (-∞,∞).