Question

For real \( x \), let \( f(x) = x^3 + 5x + 1 \), then:

• A. f is one-one but not onto R
• B. f is onto R but not one-one
• C. f is one-one and onto R
• D. f is neither one-one nor onto R

Hint:

For polynomial functions of degree 3 or higher, verify the one-one property by checking for monotonicity (increasing or decreasing) using derivatives.

Answer 1

The Correct Option is C

The given function \( f(x) = x^3 + 5x + 1 \) is a cubic polynomial. A cubic function is one-to-one and onto because it has a continuous, monotonic nature and passes through all real numbers. Therefore, \( f(x) \) is both one-one and onto.