Question: Test whether the following function is one to one, onto or both.
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$f : z \rightarrow z, f(x) = x^2 + x + 1$ **Solution:** 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 • 30 views