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One - One Function(injective) - (Concept)
A function $f: X \rightarrow Y$ is called a one-one (or injective) function, if different elements of $X$ have different images in B. i.e. no two elements of set $X$ can have the same image.
Consider,
$f: X \rightarrow Y$, function given by $y=f(x)=x$, and
$X=\{-2,2,4,6\}$ and $Y=\{-2,2,4,6\}$,
Graphically it can be shown that for every x , there is a unique y (or no y has more than one x corresponding to it) as below and hence it is one-one.
Now, consider, $\mathrm{X} 1=\{1,2,3\}$ and $\mathrm{X} 2=\{\mathrm{x}, \mathrm{y}, \mathrm{z}\}$
$
\mathrm{f}: \mathrm{X} 1 \longrightarrow \mathrm{X} 2
$
Method to check One-One Function
If $\mathrm{x}_1, \mathrm{x}_2 \in \mathrm{X}$, then $\mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{f}\left(\mathrm{x}_2\right) \Rightarrow \mathrm{x}_1=\mathrm{x}_2$
A function is one - one iff no line parallel to the X -axis meets the graph of the function at more than one point.
Even degree polynomials are NOT one-one functions
Number of One-One Function
If A and B are finite sets having elements m and n respectively, then the number of one-one functions from A to B is
$=\left\{\begin{array}{cl}{ }^n P_m & \text { if } n \geq m \\ 0 & \text { if } n<m\end{array}\right.$
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
The function defined as
, is
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Which of the following functions are one - one functions?
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Which of the following functions are injective functions ?
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Which of the following functions are one-one functions?
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Let a function $f:(0, \infty) \rightarrow[0, \infty)$ be defined by $f(x)=\left|1-\frac{1}{x}\right|$. Then $f$ is:
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Let R : x is not a positive integer. Define a function
as
then f is:
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Which of the functions f(x) will be one-one functions if is given
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Which of the following functions f(x) will be a one-one function?
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Which of the following nature of functions will be of one-one function?
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Let x denote the total number of one - one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one - one functions from set A to set A B. Then :
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If , then find the number of one-one functions from A to B.
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If and
, then find the number of one-one functions from A to B.
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If , then find the number of one-one functions from A to B.
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For real x, let then
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The number of one-one functions such that
is ___________.
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Let $f: R \rightarrow R\ {\text {be any function and }} g(x)=\frac{1}{f(x)}$. Then g is
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Let be any function. Also
is defined by
for all x. Then g is
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Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}^2+2 \mathrm{x}+1}{\mathrm{x}^2+1}$. Then
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A function $f$ from the set of natural numbers to integers defined by $f(n)=\left\{\begin{array}{l}\frac{n-1}{2}, \text { when } n \text { is odd } \\ -\frac{n}{2}, \text { when } n \text { is even }\end{array}\right.$ is
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Find the number of one-one functions from $A$ to $B$, where $n(A)=5$ and $n(B)=4$.
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Which of the following is a one-one function?
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Let f be a function from R to R given by $f\left ( x \right )=2x+\left | cosx \right |$ then f is?
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For the function $f(x)=|x-2|_{\text {and }} g(x)=4^x$
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Let $[t]$ be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$ The number of one-to-one functions from $\mathrm{A}$ to the range of $f$ is :
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If the set A contains 5 elements and set B contains 6 elements, then the number of one-one and onto mapping from A to B is
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Let $\mathrm{A}=\{(\mathrm{x}, \mathrm{y}): 2 \mathrm{x}+3 \mathrm{y}=23, \mathrm{x}, \mathrm{y} \in \mathrm{N}\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $\mathrm{A}$ to $\mathrm{B}$ is equal to $\qquad$
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The function $f: R \rightarrow[-1,1]_{\text {defined by }} f(x)=\cos x_{\text {is }}$
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The function $f(x)=x^{100}-x^{35}+59 x^{2}-1055$ is
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The function $f\left ( x \right )=x^{2}$ is
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The function $y=2x-9$ is
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Let $A$ and $B$ be two sets having elements 4 and 5 respectively. If $x$ is the number of one-one functions from $A$ to $B$ and $y$ is the number of in-to-function $B$ to $A$. Then the value of $y-x$ is
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Which of the following is a graph of a one-one function
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Let $f:[0,1] \rightarrow[-1,1]$ and $g:[-1,1] \rightarrow[0,2]$ be two functions such that $g$ is injective and $g$ o $f:[0,1] \rightarrow[0,2]$ is surjective. Then
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Let $f:[0,1] \rightarrow \mathbb{R}$ be an injective continuous function that satisfies the condition $-1<f(0)<f(1)<1$. Then the number of functions $g:[-1,1] \rightarrow[0,1]$ such that $(g \circ f)(x)=x$ for all $x \in[0,1]$ is
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If gof is a bijective function then,
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The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ is:
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Consider the sets $\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2=25\right\}$, $\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+9 \mathrm{y}^2=144\right\}, \mathrm{C}=\{(\mathrm{x}, \mathrm{y})$ $\left.\in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$, and $D=A \cap B$. The total number of one-to-one functions from the set D to the set C is:
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Definition 5 A function f : X Y is defined to be one-one (or injective), if the images
of distinct elements of X under f are distinct, i.e., for every x1, x2 X, f (x1) = f (x2)
implies x1 = x2. Otherwise, f is called many-one.
