First slide

Introduction to real valued functions

Question

The range of $\frac{1}{1 + x^{2}}$ is

Moderate

A

(0, 1]
B

(0, 2]
C

[1, 5]
D

(1, 5)

Solution

$\frac{1}{1 + x^{2}} = y \Rightarrow x = \sqrt{\frac{1 - y}{y}}$
            $\frac{1 - y}{y} \geq 0$ & $y \neq 0$

            $y (y - 1) \leq 0 \Rightarrow y \in [0, 1]$ & $y \neq 0$

                $y \in (0, 1]$

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