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Question:

For real x, let f(x) = x³ + 5x + 1, then f is one-one but not onto ℝ, f is onto ℝ but not one-one, f is neither one-one nor onto ℝ, f is one-one and onto ℝ

f is one-one and onto ℝ

f is neither one-one nor onto ℝ

f is one-one but not onto ℝ

f is onto ℝ but not one-one

Solution:

f(x) = x³ + 5x + 1
f'(x) = 3x² + 5 > 0 ⇒ f is one-one .
•Inverse of f is possible in the domain ℝ. So, f is cubic ⇒ f is onto
f is one-one and onto.