Question

Which of the following function is surjective but not injective

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

Answer 1

Answer: (D)

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

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