Final answer:
To determine if a set of functions is linearly independent on the interval (-∞, ∞), we check if the only solution to the equation ax + by + cz = 0 is a = b = c = 0. By plugging in specific values of x, we can show that a = b = c = 0 is the only solution.
Step-by-step explanation:
To determine whether the given set of functions is linearly independent on the interval (-∞, ∞), we need to check if the only solution to the equation a*f₁(x) + b*f₂(x) + c*f₃(x) = 0, for all x in (-∞, ∞), is a = b = c = 0. Let's assume a, b, and c are constants.
Plugging in the given functions, we have a*5 + b*cos²x + c*sin²x = 0. To show that a = b = c = 0 is the only solution, we can choose specific values of x that yield simple equations. For example, when x = 0, we get a*5 + b*1 + c*0 = 0, which simplifies to a + b = 0. When x = π/4, we get a*5 + b*½ + c*½ = 0, which simplifies to a + b + c = 0. By solving these two equations simultaneously, we find that a = b = c = 0, confirming that the given set of functions is linearly independent on the interval (-∞, ∞).