Question

let :R➡️R be defined as f(x) =2x-1/3, prove that f is bijection of f(x) and hence find inverse of x

Answer 1

your answer for above question

Answer 2

F is a bijection of f(x) and $f^{-1}(x) = \frac{6x+2}{12}$

Given:

Function F: R to R, defined as F(x)=2x-1/3

To Find:

F is a bijective function of f(x) and hence to find the inverse of x.

Solution:

$If\\f(x_{1} )=f(x_{2})\\ 2x_{1}-\frac{1}{3} = 2x_{2}-\frac{1}{3} \\ 2x_{1}= 2x_{2}\\ x_{1}= x_{2}$

Therefore, F(x) is one-one.

Let f(x)=y

$y=2x-\frac{1}{3}\\ 2x=y +\frac{1}{3}\\ x=\frac{y}{2} +\frac{1}{6}\\ x=\frac{6y+2}{12}-------------(1)$

X belongs to R.

Therefore, there exists at least one  Y such that f(x)=y for all X that belongs to R.

Hence F(x) is Onto.

F(x) is both One-One and Onto. Hence, F is a bijection of F(x) and F(x) is invertible.

Interchange x and y in equation (1)

$y=\frac{6x+2}{12}$

$f(y)=x\\\\f^{-1}(x)=y\\\\f^{-1}(x) = \frac{6x+2}{12}$

∴ F is a bijection of f(x) and $f^{-1}(x) = \frac{6x+2}{12}$

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