Answer
Verified
477.6k+ 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
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE