Question

Which of the following function is surjective but not injective

• A. $f:R\to R,f(x)=x^4+2x^3-x^2+1$
• B. $f:R\to R,f(x)=x^3+x+1$
• C. $f:R\to R^{+},f(x)=\sqrt{1+x^2}$
• D. $f:R\to R,f(x)=x^3+2x^2-x+1$

Answer 1

Answer: (D)

(A)$f(x)=x^4+2x^3-x^2+1\to$ A polynomial of degree even will always be into
say
$f(x)=a_0{x}^{2n}+a_1{x}^{2n-1}+a_2{x}^{2n-2}+\dots .+a_{2n}$

Hence it will never approach $\infty /-\infty$
(B) $f(x)=x^3+x+1\Rightarrow f^{\prime }(x)=3x^2+1\Rightarrow$ injective as well as surjective
(C) $f(x)=\sqrt{1+x^2}$ - neither injective nor surjective (minimum value $=1$ )
$f(x)=x^3+2x^2-x+1\Rightarrow f^{\prime }(x)=3x^2+4x-1\Rightarrow D>0$
Hence $f(x)$ is surjective but not injective.]