sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions, Then the function

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Solution

Q.
Let $f, g : R \to R$ be defined as: [2024]

$f (x) = | x - 1 |$ and $g (x) =$ ${\begin{matrix} e^{x}, & x \geq 0 \\ x + 1, & x \leq 0 \end{matrix} .$

Then the function $f (g (x))$ is

1 onto but not one-one.

2 neither one-one nor onto.

3 both one-one and onto.

4 one-one but not onto.

Ans.
(2)

$f (g (x)) = | g (x) - 1 |$

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

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

$f (g (x))$ is neither one-one nor onto as negative numbers have no pre-image.

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