Question

How do you find the inverse of $f(x)=\dfrac{2x+1}{x-3}$ ?

Answer 1

Hint: In this question, we have to find ${{f}^{1}}(x)$ i.e. inverse of a function. Here compositions of functions are used. The best way to find the inverse function is by finding y = f(x), making x the subject and then switching y and x. Main point is to initialize $\dfrac{2x+1}{x-3}$ with ‘y’.

Complete step by step answer:
Let’s solve the question now.
You need to understand first, what a composition of functions is. Basically it means to combine two or more functions together in such a manner that the output from one function becomes the input for the next function.
There are some properties of composition of functions. Let’s discuss them.
First one is associative property which says that if there are given three functions i.e. f, g, and h then,
\[f\circ (g\circ h)\text{ }=\text{ }(f\circ g)\circ h\].
Second one is commutative property which says that two functions f and g commute each other, then,
$f\circ g=g\circ f$
Now, let’s come to the question.
$\Rightarrow f(x)=\dfrac{2x+1}{x-3}$
We will follow the easiest way to find the inverse function. For that, first let y = f(x).
$\Rightarrow y=\dfrac{2x+1}{x-3}$
Next step is to take the denominator to the other side. We get:
$\Rightarrow y\left( x-3 \right)=2x+1$
Now, the next step is to open the bracket and multiply the terms with ‘y’. We get:
 $\Rightarrow yx-3y=2x+1$
Take like terms on one side:
$\Rightarrow xy-2x=3y+1$
Take ‘x’ common from xy – 2x. We get:
$\Rightarrow x\left( y-2 \right)=3y+1$
Take (y – 2) to the other side of the equation. We get:
$\Rightarrow x=\dfrac{3y+1}{y-2}$
Now, for finding ${{f}^{-1}}$(x), place y = x in the above equation. We get:
$\Rightarrow {{f}^{-1}}(x)=\dfrac{3x+1}{x-2}$
So, this is our final answer.

Note: Mistake can be done at the final step when x is not written at the place of y. So be careful at that step. In inverse functions, questions can come in algebraic form, quadratic equation form and many more. For these forms you should know all the properties of composition of functions.