Question

A function $f:A\to B$, where $A=\{x:-1\le x\le 1\}$ and $B=\{y:1\le y\le 2\}$ is defined by the rule $y=f\left(x\right)=1+x^2$. Which of the following statement is true?

• A. f is injective but not surjective
• B. f is surjective but not injective
• C. f is both injective and surjective
• D. f is neither injective nor surjective

Answer 1

Answer: (B)

f is surjective but not injective

$A=\left\{x:-1\le x\le 1\right\}$ and $B=\left\{y:1\le y\le 2\right\}$.

For $x=-1$,

$y=f\left(-1\right)$

$=1+{\left(-1\right)}^2$

$=2$

For $x=1$,

$y=f\left(1\right)$

$=1+{\left(1\right)}^2$

$=2$

Therefore, $f\left(-1\right)=f\left(1\right)$ but $-1\ne 1$, so the function is not injective.

Since, for every B, there is a preimage, therefore, the function is surjective.