Question

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=x^3-3x+1 \] Then the function \(f\) is:

• A. One-one and onto
• B. One-one but not onto
• C. Onto but not one-one
• D. Neither one-one nor onto

Hint:

For polynomial functions, use derivatives to test one-one nature and end behavior to test onto nature.

Answer 1

The Correct Option is C

Concept: To determine whether a function is one-one or onto:
• A function is one-one (injective) if different inputs always produce different outputs.
• A function is onto (surjective) if every real number in the codomain has at least one pre-image. For polynomial functions:
• Odd degree polynomials with real coefficients are generally onto from \(\mathbb{R}\to\mathbb{R}\).
• To test one-one nature, examine monotonicity using derivatives.

Step 1: Check whether the function is onto. Given: \[ f(x)=x^3-3x+1 \] This is a cubic polynomial. Observe the behavior as \(x\to\infty\): \[ f(x)\to\infty \] and as \(x\to-\infty\): \[ f(x)\to-\infty \] Since the function is continuous and takes all real values from \(-\infty\) to \(+\infty\), every real number has a pre-image. Hence the function is onto.

Step 2: Check whether the function is one-one. Differentiate: \[ f'(x)=3x^2-3 \] \[ =3(x^2-1) \] \[ =3(x-1)(x+1) \] Critical points occur at: \[ x=-1,\qquad x=1 \] Now examine sign changes. \[ f'(x)>0 \quad \text{for} \quad x<-1 \] \[ f'(x)<0 \quad \text{for} \quad -1<x<1 \] \[ f'(x)>0 \quad \text{for} \quad x>1 \] Thus, the function first increases, then decreases, then increases again. Therefore the function is not strictly monotonic on \(\mathbb{R}\). Hence it is not one-one.

Step 3: Conclude the result. We found:
• Function is onto
• Function is not one-one Therefore, \[ \boxed{ \text{Onto but not one-one} } \] Hence the correct answer is: \[ \boxed{(C)} \]