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18001022727Function as we already know is basically a machine and we have different kinds of machine in this universe , so is the case with functions also .Functions can be classified on the basis of their domain, codomain & range.Let’s dig into the different types of functions in detail.
Table of Contents
A function f : XY is defined to be oneone (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x_{1}, x_{2}X, f(x_{1})=f(x_{2}) x_{1}=x_{2} .
Example: Let us consider the two functions as shown below:
In both the mappings, every element of set A is related to only one element of set B and no two elements in set A are related to the same element in set B. Therefore, both the mappings are injective or oneone.
There are two methods to determine whether the given function is oneone or not:
Method 1: Algebraic/Analytical Method
Let f:AB be a function, x_{1 }and x_{2} be the two elements in set A, and the corresponding images of x_{1} and x_{2} in set B be f(x_{1}) and f(x_{2}), respectively. Now, for f to be a oneone function, f(x_{1})=f(x_{2})x_{1}=x_{2} or x_{1}x_{2}f(x_{1})f(x_{2})
Example: Consider the function f:ℝℝ such that f(x)=3x+5.Now, in order to prove it is oneone, Let x_{1}, x_{2}ℝ such that f(x_{1})=f(x_{2})
3x_{1}+5=3x2+5
3x_{1}=3x2
x_{1}=x2
So, x_{1}, x_{2}X, f(x_{1})=f(x_{2}) x_{1}=x_{2}
Hence,the images of distinct elements of X under f are distinct.Therefore, the given function is oneone.
Example: Consider the function f:ℝℝ such that f(x)=x^{2}+2.Now, let's check whether it is oneone or not, Let x_{1}, x_{2}ℝ such that f(x_{1})=f(x_{2})
x^{1}_{2}+2=x^{2}_{2}+2
x^{1}_{2}=x^{2}_{2}
x_{1}=x_{2} or x_{1}=x_{2}
So, x_{1}, x_{2}X, f(x_{1})=f(x_{2}) x_{1}=x_{2}
Hence, the images of distinct elements of X under f are not distinct.Therefore, the given function is not oneone.
Method 2: Graphical Method (Horizontal line test)
Firstly, draw a horizontal line( parallel to xaxis) upon the graph of the function. Then,
This is just the graphical interpretation of ‘if xy, then f(x)f(y)
Example: Consider the function f(x)=y=x^{3}
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When we draw a horizontal line, parallel to xaxis, it cuts the graph of only at one point, hence the function is oneone.
Method 3: Derivative Test
Let f:AB be a continuous and differentiable function, then f is oneone function if f'(x)<0 or f'(x)>0.
Example: Consider the function f:ℝ{2}ℝ{1} such that Check whether the function is an injection or not.
Solution If f(x) is continuous and differentiable function then it will be injection if f'(x)<0 or f'(x)>0.
Here f(x) is a rational function, and a rational function is continuous and differentiable over it’s entire domain.
Therefore, f is oneone.
For a function f: AB, if n(A)=m and n(B)=n, then the number of oneone functions would be:
Example: The number of oneone functions from set P={1,2,3} to set Q={4,5,6,7} is^{4}P_{3} as n(A)=m= 3 and n(B)=n= 4.
Therefore, number of oneone functions from set P to Q is 24.
A function f:AB is said to be a manyone if there exist at least two or more elements of set A that have the same image in set B.In other words a function which is not oneone is manyone.
Example: Consider the given mapping,
It represents a manyone function as image of both x_{1},x_{2}A is the same, i.e., y_{1}B. Or we can write it as f(x_{1})=f(x_{2})=y_{1}.
Note:
Example: Consider the graph of the function f(x)=y=x^{2}4x+3=(x2)^{2}1
When we draw an intersecting horizontal line parallel to xaxis, it intersects the graph of f(x) at more than one point as shown in the figure below. So, different values of x have the same image on y. Therefore, it is a manyone function
Number of manyone functions = (Total number of functions)  (number of oneone functions)
In general, if f:AB, n(A)=m and n(B)=n, then the number of manyone functions would be =n^{m}  ^{n}P_{m}.
Example: If A={1,2,3,4}, then the number of functions on set A, which are manyone, is:
Solution
Number of manyone functions = (Total number of functions)  (number of oneone functions) Number of manyone functions =n^{m}  ^{n}P_{m};
Here, n=m=4;
Number of manyone functions= 4^{4}  ^{4}P_{4}=256  24=232.
Hence, option (c) is the correct answer.
If f:A→ B, is a function such that each and every element in B(codomain) has at least one preimage in A, then we say that f is an onto function.
In other words, iff b ∊ B, there exists some a ∈ A such that f(a)=b, then f is an onto function. An onto function is also known as a surjective function.
Note: A function, f is said to be an onto function if the range of f= codomain of the function f.
Example:
Both the functions given above are onto functions, as every element in the codomain set has at least one preimage in the domain set.
Number of onto functions
For obtaining the formula for the number of onto functions let us have a quick recap of the “Principle of inclusion & exclusion“
For two sets A & B the principle of inclusion and exclusion can be understood as:
Similarly for three sets A, B & C the principle of inclusion and exclusion can be understood as:
For a function f: AB, if n(A)=m and n(B)=n, then
Case 1: When m n
Number of onto functions = Total number of functionsTotal number of functions where at least one element is excluded
Number of onto functions
Case 2: When n>m, onto function doesn’t exist.
Related concept video: https://www.youtube.com/watch?v=cGUyOykwvk&t=340s
If a function, f:A→ B, is such that there exists at least one element in B (codomain) that is not the image of any element of A, then function f is known as an into function.
In other words , if the range of the function is a proper subset of its codomain, then the function is said to be an into function. Range Codomain.
Example: Consider the mapping given below.
The function represented by this mapping is an into function as the elements a,e,fX2(codomain) are not the image of any element of set X1(domain).
Note: Number of into functions = (Total number of functions)  (Number of onto functions).
Example: f: ℝ ℝ be a function such that f(x) =x^{2}+x2, Check whether it is an into function or not?
Solution: Plotting the graph for f(x) =x^{2}+x2
Here, the domain and codomain of the given function is ℝ and the range of the function from the graph is [94,∞).As the range is a subset of the codomain, the given function is an into function.
Example: Identify f(x) = xx as a oneone or manyone function.
Solution:Given, f(x) = xx
The graph for f(x) is plotted below:
Any line drawn parallel to xaxis will cut the graph at only one point.Therefore, the given function is a oneone function.
Example: Determine whether the function f(x) =x , where f: ℝ^{+} ℝ is into or onto function.
Solution
f(x) =x , where x ∊ ℝ^{+} , For x ∊ ℝ^{+}, f(x) = x. Now plotting the graph for f(x)
For the domain, x ∊ ℝ+, the range of the function is y ∊ ℝ+and the codomain of the function is given as ℝ. Since the range and the codomain of the given function are not equal, the given function is an into function.
Example: Determine whether the function f(x) given by is oneone or not.
Solution:
As f'(x)>0, therefore f(x) is a oneone function.
Example: Determine whether the function f: ℝ{0} ℝ, defined by f(x)=1logx is an onto function or not.
Solution:
Let’s draw the graph of the function first.
As given, the domain of the function is ℝ{0} and the codomain is ℝ.
Clearly from the graph, the range of the function is (,1]. So, range codomain.
Therefore, the function is not onto.
Q 1 . Can a function be both one one as well as onto?
Answer: Yes a function can be both one one and onto simultaneously. Such functions are known as bijective functions.
Q 2. What is the difference between Vertical and Horizontal line test?
Answer: A vertical line test is used to check whether a relation is a function or not whereas a horizontal line test is used to check whether a function is one one or not.
Q 3.How to determine whether a function is onto using graph?
Answer: Using graph, we can figure out the range of the function and then we can compare it with the codomain to predict whether the given function is onto or not.
Q 4. Are oneone functions invertible?
Answer: For a function to be invertible, it should be both one one as well as onto.So we cannot predict whether the function is invertible just on the basis of one one condition.
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