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# State with reason whether the following functions have inverse i.$f:\left\{ {1,2,3,4} \right\} \to \{ 10\} with \\ f = \{ (1,10),(2,10),(3,10),(4,10)\} \\$ii.$g:\left\{ {5,6,7,8} \right\} \to \{ 1,2,3,4\} with \\ g = \{ (5,4),(6,3),(7,4),(8,2)\} \\$iii.$h:\left\{ {2,3,4,5} \right\} \to \{ 7,9,11,13\} with \\ h = \{ (2,7),(3,9),(4,11),(5,13)\} \\$  If the given functions satisfy both one-one and onto functions then it will have
inverse.

Given,
$f:\left\{ {1,2,3,4} \right\} \to \{ 10\} with \\ f = \{ (1,10),(2,10),(3,10),(4,10)\} \\$
Here, the domain of ‘f’ is $\{ 1,2,3,4\}$and co-domain is$\{ 10\}$.
As we know a function is said to be a one-one function if distinct elements of domain
mapped with distinct elements of co-domain.
$f = \{ (1,10),(2,10),(3,10),(4,10)\}$
But, in this case if we see the function ‘f’ each element from the domain is mapped with the
same element from co-domain i.e.., 10.Since, all the elements have the same image 10 which is not satisfying the one-one function condition. Hence, f is not a one-one function.
Therefore, f doesn’t have inverse.

ii.Given,
$g:\left\{ {5,6,7,8} \right\} \to \{ 1,2,3,4\} with \\ g = \{ (5,4),(6,3),(7,4),(8,2)\} \\$
Here, the domain of ‘g’ is $\{ 5,6,7,8\}$and co-domain is$\{ 1,2,3,4\}$.
As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.
$g = \{ (5,4),(6,3),(7,4),(8,2)\}$
But, in this case if we see the function ‘g’, the elements 5 and 7 from the domain are mapped
with the same element from co-domain i.e... As ‘g’ is not satisfying the one-one function
condition. Hence,’g’ is not a one-one function.
Therefore, g doesn’t have inverse.

iii.Given,
$h:\left\{ {2,3,4,5} \right\} \to \{ 7,9,11,13\} with \\ h = \{ (2,7),(3,9),(4,11),(5,13)\} \\$
Here, the domain of ‘h’ is $\{ 2,3,4,5\}$and co-domain is$\{ 7,9,11,13\}$.
As we know a function is said to be a one-one function if distinct elements of domain mapped with distinct elements of co-domain.
$h = \{ (2,7),(3,9),(4,11),(5,13)\}$

Here, each element from the domain is mapped with the different element from the co-domain. Therefore, ‘h’ is a one-one function.
Now, let us check with the onto condition i.e.., each element in the co-domain has a pre-image from the domain.
Here, each element from the co-domain has a pre-image from the domain. Therefore ‘h’ is
an onto function. As, function ‘h’ is both one-one and onto functions.
Hence, the inverse of ‘h’ exists i.e..,
$h = \{ (2,7),(3,9),(4,11),(5,13)\} \\ {h^{ - 1}} = \{ (7,2),(9,3),(11,4),(13,5)\} \\$
Hence, among the functions ‘f’, ‘g’, ‘h’ only the function ‘h’ has the inverse.

Note: The alternate method to find whether a function is one-one is by horizontal line test
i.e. ., if a horizontal line intersects the original function in a single region, the function is a
one-to-one function otherwise it is not a one-one function.

View Notes
Introduction to Composition of Functions and Find Inverse of a Function  Introduction to the Composition of Functions and Inverse of a Function  Composition of Functions and Inverse of a Function  Inverse Functions  Inverse Trigonometric Functions  Important Properties of Inverse Trigonometric Functions  Graphical Representation of Inverse Trigonometric Functions  How to Find The Median?  The Solid State  Preposition of Cause Reason Purpose  