Final answer:
The expression f(-5/a²) for the given function f(x) = (-2x²)/(-x+5x⁻¹) simplifies to -50/(a⁴ + 5a²), provided that a ≠ 0.
Step-by-step explanation:
To solve for f(-5/a²) when given the function f(x) = (-2x²)/(-x+5x⁻¹), begin by substituting x with -5/a². Here's the process:
f(-5/a²) = (-2(-5/a²)²)/(-(-5/a²) + 5(-5/a²)⁻¹)
Simplify the numerator by squaring -5/a²:
f(-5/a²) = (-2(25/a⁴))/(-(-5/a²) + 5(-a²/5))
Then simplify the denominator:
f(-5/a²) = (-50/a⁴)/(5/a² + a²)
Multiplying the fraction by a⁴/a⁴ to clear the denominator:
f(-5/a²) = (-50)/(5a² + a⁴)
Finally, simplify to get the answer:
f(-5/a²) = -50/(a⁴ + 5a²)
Note that we must assume a ≠ 0 to avoid division by zero. Also, the original expression must be valid for the substitution; in particular, the denominator must not be zero, which implies a² ≠ 5 for the solution to hold.