The end behavior of the function is as x→-∞, f(x) → ∞ and x → ∞, f(x) → -∞
The function is given as:
f(x)=2x^7 − 5x^3 − 2x + 1
As x → −∞, we set x = −∞
So, we have:
f(−∞) = 2(−∞)^7 − 5(−∞)^3 − 2(−∞) + 1
Evaluate the exponents and the products
f(−∞) = −∞ + ∞ + ∞ + 1
Evaluate the sum and the difference
f(−∞) = ∞
This means that as x→-∞, f(x) → ∞
As x→∞, f(x)→ we set x = ∞
So, we have:
f(∞) = 2(∞)^7 − 5(∞)^3 − 2(∞) + 1
Evaluate the exponents and the products
f(∞) = ∞ - ∞ - ∞ + 1
Evaluate the sum and the difference
f(∞) = -∞
This means that as x → ∞, f(x) → -∞
Hence, the end behavior of the function is as x→-∞, f(x) → ∞ and x → ∞, f(x) → -∞
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