Question

For real $\mathrm{x},\mathrm{if}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+5\mathrm{x}+1,$ 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 $R$

Answer 1

The correct option is C

$f$is one-one and onto $R$

The explanation for the correct option:

Step1. Find whether the function is one-one or not:

$$\begin{gathered} f\left(x\right)=x^3+5x+1 \\ \therefore f^{\text{'}}\left(x\right)=3x^2+5\left[\mathbf{\because }\frac{\mathbf{d}\mathbf{\left(}{\mathbf{x}}^{\mathbf{n}}\mathbf{\right)}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{n}{\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\right] \\ {f}^{\text{'}}\left(x\right)>0\mathbf{[}\mathbf{\because }\mathbf{3}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathbf{>}\mathbf{0}\mathbf{}\mathbf{,}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\in }\mathbf{R}\mathbf{}\mathbf{]} \end{gathered}$$

$\forall x\in R,f\left(x\right)$ is strictly increasing function.

$\therefore \forall x\in R,f\left(x\right)$ is a one-one function,

Step2. Find whether the function is onto or not:

$f\left(x\right)$ is polynomial, continuous, and increasing function.

$\begin{array}{rcl}\forall y\exists x& \in & R\\ f\left(x\right)& =& y\left[\because f\left(x\right)isapolynomialofodddegree\forall y\left(codomain\right)\exists x\in R\right]\\ \underset{x\to +\infty }{\lim }f\left(x\right)& =& \underset{x\to +\infty }{\lim }{x}^3+5x+1\\ & =& +\infty \\ \underset{x\to -\infty }{\lim }f\left(x\right)& =& \underset{x\to -\infty }{\lim }{x}^3+5x+1\\ & =& -\infty \end{array}$

$$\begin{gathered} \Rightarrow Rangeoff\left(x\right)=(-\infty ,+\infty ) \\ =R \\ =Codomain \end{gathered}$$

$\therefore f\left(x\right)$ is onto function.

Hence, Option(C) is the correct answer.