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State with reason whether following functions have inverses: (i) \( f: \{ 1,2,3,4,\} \to \{10\}\) with \( f= \{(1,10),(2,10),(3,10), (4,10)\}\)
This is part of a multipart question and answered separately here on Clay6.com
class12
cbse
ch1
sec3
q5
q5-1
easy
sec-a
p18
math
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asked
Mar 19, 2013
by
balaji.thirumalai
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Toolbox:
To check if a function is invertible or not ,we see if the function is both one-one and onto.
A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function.
A function$ f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given $f:\{1,2,3,4\} \to \{10\}$ with $f=\{(1,10),(2,10,(3,10)(4,10)\}$
Step 1: Checking for one-one:
From the given definition of $f$, we can see that it is many-one, as for every ${1,2,3,4} \in f, f(1) = f(2) = f(3) = f(4) = 10$.
Therfore, the function does not have an inverse. We don't need to check further for onto.
answered
Mar 19, 2013
by
balaji.thirumalai
State with reason whether following functions have inverse: (ii) \(g: \{ 5,6,7,8\} \to \{1,2,3,4\}\) with \(g= \{(5,4),(6,3),(7,4),(8,2)\}\)
True or False: The function $f$ has an inverse: $ f:{1,2,3,4,}\to{10}$ with $f={(1,10),(2,10),(3,10),(4,10)}$
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