Let and be defined as: and Then, is, sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions, be defined as- 2

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Solution

Q.
Let $f : R \to R$ and $g : R \to R$ be defined as:

$f (x)$ $= {\begin{matrix} \log_{e} x, & x > 0 \\ e^{- x}, & x \leq 0 \end{matrix}$ and $g (x)$ $= {\begin{matrix} x, & x \geq 0 \\ e^{x}, & x < 0 \end{matrix} .$

Then, $g o f : R \to R$ is [2024]

1 neither one-one nor onto

2 onto but not one-one

3 one-one but not onto

4 both one-one and onto

Ans.
(1)

$g (f (x))$ $=$ ${\begin{matrix} f (x), & f (x) \geq 0 \\ e^{f (x)}, & f (x) < 0 \end{matrix}$

$=$ ${\begin{matrix} \log_{e} x, & x \geq 1 \\ e^{- x}, & x \leq 0 \\ e^{\log_{e} x} = x, & 0 < x < 1 \end{matrix}$

$g o f (x) =$ ${\begin{matrix} e^{- x}, & x \leq 0 \\ x, & 0 < x < 1 \\ \log_{e} x, & x \geq 1 \end{matrix}$

Now, range of $g o f = [0, \infty) \neq R$

Hence, not one-one and not onto.

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