Let $ f : R $ - { $ \frac {5}{4} $ } $ \rightarrow R $ be a function defined as $ f(x) = \frac{5x}{4x+5} $ . The inverse of $ f $ is the map $ g : Range\,f\,\rightarrow R $ - { $ \frac {5}{4} $ } given by
- $ g(y) = \frac{y}{5-4y} $
- $ g(y) = \frac{y}{5+4y} $
- $ g(y) = \frac{5y}{5-4y} $
- $None\, of\, these$
The Correct Option is C
Solution and Explanation
Let $y=\frac{5x}{4x+5}$
$\Rightarrow 4xy+5y=5x$
$\Rightarrow x\left(5-4y\right)=5y$
$\Rightarrow x=\frac{5y}{\left(5-4y\right)}=f^{-1}\left(y\right)$
$\Rightarrow f^{-1}\left(x\right)=\frac{5x}{5-4x} $
or $g\left(y\right)=\frac{5y}{5-4y}$
which is the inverse of $f$ is the map g.
Range $f \rightarrow R-\left\{\frac{5}{4}\right\}$
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions