Answer

Verified

449.1k+ views

Hint: To find an inverse function such as f(x) we have the following method: Express x in terms of f(x) and then replace x with g(x) and f(x) with x. The resultant function g(x) will be the inverse of the function f(x) then we check which of the options are correct.

“Complete step-by-step answer:”

We have the function $f(x)=\dfrac{3x+2}{5x-3}$ . First of all let us express x in terms of f(x). For that we have,

$f(x)[5x-3]=3x+2$

Multiplying f(x) we have,

$5xf(x)-3f(x)=3x+2$

Taking 3x in LHS and -3f(x) in RHS we have,

$5xf(x)-3x=3f(x)+2$

Taking x common from the terms in LHS we have,

$x(5f(x)-3)=3f(x)+2$

Dividing both sides with coefficient of x we have,

$x=\dfrac{3f(x)+2}{5f(x)-3}$

Now replacing x with g(x) and f(x) with x we have,

$g(x)=\dfrac{3x+2}{5x-3}$

This function g(x) is the inverse of the function f(x). Hence, we can write ${{f}^{-1}}(x)=\dfrac{3x+2}{5x-3}$ .

We had $f(x)=\dfrac{3x+2}{5x-3}$ and ${{f}^{-1}}(x)=\dfrac{3x+2}{5x-3}$ . Therefore $f(x)={{f}^{-1}}(x)$ .

Hence, option A is the correct answer.

Note: We should know that the inverse of a function is a mirror image of the function about the line $y=x$ means if we were to plot the graph of a function and its inverse we will find that they are mirror image of each other about the line $y=x$ .

This the graph of the function $f(x)=\dfrac{3x+2}{5x-3}$ . As we can see the function is perfectly symmetric and if we were to draw the mirror image it would again give the same function.

“Complete step-by-step answer:”

We have the function $f(x)=\dfrac{3x+2}{5x-3}$ . First of all let us express x in terms of f(x). For that we have,

$f(x)[5x-3]=3x+2$

Multiplying f(x) we have,

$5xf(x)-3f(x)=3x+2$

Taking 3x in LHS and -3f(x) in RHS we have,

$5xf(x)-3x=3f(x)+2$

Taking x common from the terms in LHS we have,

$x(5f(x)-3)=3f(x)+2$

Dividing both sides with coefficient of x we have,

$x=\dfrac{3f(x)+2}{5f(x)-3}$

Now replacing x with g(x) and f(x) with x we have,

$g(x)=\dfrac{3x+2}{5x-3}$

This function g(x) is the inverse of the function f(x). Hence, we can write ${{f}^{-1}}(x)=\dfrac{3x+2}{5x-3}$ .

We had $f(x)=\dfrac{3x+2}{5x-3}$ and ${{f}^{-1}}(x)=\dfrac{3x+2}{5x-3}$ . Therefore $f(x)={{f}^{-1}}(x)$ .

Hence, option A is the correct answer.

Note: We should know that the inverse of a function is a mirror image of the function about the line $y=x$ means if we were to plot the graph of a function and its inverse we will find that they are mirror image of each other about the line $y=x$ .

This the graph of the function $f(x)=\dfrac{3x+2}{5x-3}$ . As we can see the function is perfectly symmetric and if we were to draw the mirror image it would again give the same function.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE