f: R → R defined by f(x) = 1 + x2 - Vidyadip

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f: R → R defined by f(x) = 1 + x²

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Answer

f: R → R is defined as
f(x) = 1 + x²
Let $\text{x}_1,\text{x}_2\in\text{R}$ such that f(x₁) = f(x₂)
$\Rightarrow1+\text{x}_{1}^{2}=1+\text{x}_{2}^{2}$
$\Rightarrow\text{x}_{1}^{2}=\text{x}_{2}^{2}$
$\Rightarrow\text{x}_{1}=\pm\text{x}_{2}$
$\therefore$ f(x₁) = f(x₂) does not imply that x₁ = x₂.
For instance,
f(1) = f(-1) = 2
$\therefore$ f is not one-one.
Consider an element -2 in co-domain R.
It is seen that f(x) = 1 + x² is positive for all $\text{x}\in\text{R}.$
Thus, there does not exist any x in domain R such that f(x) = -2.
$\therefore$ f is not onto.
Hence, f is neither one-one nor onto.

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