Test whether the following function is one to one, onto or both.


$f : z \rightarrow z, f(x) = x^2 + x + 1$


Let function f : z $\rightarrow$ z defined as

$f(x) = x^2 + x + 1$

1] Injective or one to one.

A function f : z $\rightarrow$ z is said to be an injective or one to one function if

$f(x_1) = f(x_2) \rightarrow x_1 = x_2$ OR

$f(x_1) \neq f(x_2) \rightarrow x_1 \neq x_2$

say, $f(x_1) = f(x_2)$

$x_1^2 + x_1 + 1 = x_2^2 + x_2 + 1 $

$\therefore$ $x_1^2 + x_1 \neq x_2^2 + x_2$

Any function of 2nd def is never injective or one to one.

$\therefore$ f : z $\rightarrow$ z, $f(x) = x^2 + x + 1$

is not injective function.

2] Subjective or onto function.

A function $ f : z \rightarrow z$ is said to be a subjective function if

co-domain (+) = Range (+)

OR V y E Z a pre image X W Z.

In the given function $f(x) = x^2 + x +1 $ it can be seen that negative elements of Z are not image of any element.

$\therefore$ The given function is not subjective or onto function.

renu • 224  views
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