(D) Given the function $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$.Case $I$: If $x \geq 0$, then $|x| = x$. Thus, $f(x) = \frac{e^x - e^{-x}}{e^x +

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$f$ is neither one-one nor onto

(D) Given the function $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$.Case $I$: If $x \geq 0$, then $|x| = x$. Thus, $f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \tanh(x)$. As $x$ increases from $0$ to $\infty$, $f(x)$ increases from $0$ to $1$.Case $II$: If $x Since $f(x) = 0$ for all $x Since the range of $f$ is $[0, 1)$, which is a proper subset of the codomain $R$, the function is not onto.Therefore, $f$ is neither one-one nor onto.

Class 12 Mathematics Relation and Function Type of Functions based on Mapping

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Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$, then

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A
$f$ is both one-one and onto
B
$f$ is one-one but not onto
C
$f$ is onto but not one-one
D
$f$ is neither one-one nor onto

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Relation and Function

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Type of Functions based on Mapping

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On the set of integers $Z$,define $f: Z \rightarrow Z$ as $f(n) = \begin{cases} \frac{n}{2}, & n \text{ is even} \\ 0, & n \text{ is odd} \end{cases}$. Then $f$ is:

EasyKCET 2009

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Statement-$I$: Let $f : R \rightarrow R$ be a function such that $f(x) = x^3 + x^2 + 3x + \sin x$. Then $f$ is a one-one function.
Statement-$II$: $f(x)$ is a decreasing function.

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Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: A \rightarrow B$,such that $f(1)+f(3)=14$,is:

DifficultJEE MAIN 2024

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If a function $f: R-\{l\} \to R-\{m\}$ defined by $f(x) = \frac{x+3}{x-2}$ is a bijection,then $3l - 2m =$

MediumAP EAMCET 2022

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If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} 2x & : x > 3 \\ x^2 & : 1 < x \leq 3 \\ 3x & : x \leq 1 \end{cases}$,then the value of $f(-1) + f(2) + f(4)$ is:

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Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}}$, then

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On the set of integers $Z$,define $f: Z \rightarrow Z$ as $f(n) = \begin{cases} \frac{n}{2}, & n \text{ is even} \\ 0, & n \text{ is odd} \end{cases}$. Then $f$ is:

Statement-$I$: Let $f : R \rightarrow R$ be a function such that $f(x) = x^3 + x^2 + 3x + \sin x$. Then $f$ is a one-one function.Statement-$II$: $f(x)$ is a decreasing function.

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: A \rightarrow B$,such that $f(1)+f(3)=14$,is:

If a function $f: R-\{l\} \to R-\{m\}$ defined by $f(x) = \frac{x+3}{x-2}$ is a bijection,then $3l - 2m =$

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} 2x & : x > 3 \\ x^2 & : 1 < x \leq 3 \\ 3x & : x \leq 1 \end{cases}$,then the value of $f(-1) + f(2) + f(4)$ is:

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Statement-$I$: Let $f : R \rightarrow R$ be a function such that $f(x) = x^3 + x^2 + 3x + \sin x$. Then $f$ is a one-one function.
Statement-$II$: $f(x)$ is a decreasing function.