Question

Let A = {-2, -1, 0, 1, 2} and $f:A\to Z$ be a function defined by $f\left(x\right)=x^2-2x-3$. Find :
i) Range of f i.e. f(A)
ii) Pre-images of 6, -3 and 5.

Answer 1

We have,
$f\left(x\right)=x^2-2x-3$
Now,
$f(-2)=(-2{)}^2-2(-2)-3$
$=4+4-3$
$=5$
$f(-1)=(-1{)}^2-2(-1)-3$
$=1+2-3$
$=0$
$f(-0)=(-0{)}^2-2\times 0-3$
$=-3$
$f\left(1\right)=(1{)}^2-2\times 1-3$
$=1-2-3$
$=-4$
$f\left(2\right)=(2{)}^2-2\times 2-3$
$=4-4-3$
$=-3$
(a) Range (f) = {-4,-3,0,5}
(b) Clearly, pre-images of 6, -3 and 5 is $\varphi$, {0, 2}, -2 respectively.