Final answer:
The functions f₁(x)=x, f₂(x)=x-1, and f₃(x)=x+9 are not linearly independent on the interval (-∞, ∞) because their Wronskian determinant is zero, which does not depend on x.
Step-by-step explanation:
To determine whether the set of functions f₁(x)=x, f₂(x)=x-1, and f₃(x)=x+9 is linearly independent on the interval (-∞, ∞), we can use the Wronskian determinant. For a set of functions to be linearly independent, their Wronskian determinant must not be identically zero. The Wronskian determinant is calculated as follows:
Wronskian Determinant Calculation
For our set of functions, the matrix and its determinant are:
Matrix:
| f₁(x) f₂(x) f₃(x) |
| x x-1 x+9 |
| f₁'(x) f₂'(x) f₃'(x) |
| 1 1 1 |
The determinant of this matrix (Wronskian) simplifies to:
W = | 1 1 1 |
| 1 1 1 | = 0 (the second row is a repeat of the first).
Since the determinant of the matrix is zero and does not depend on x, the set of functions is not linearly independent on the interval (-∞, ∞).