Question

Given $f(x) = \dfrac{{2x - 1}}{{x - 1}}$, how do you find ${f^{ - 1}}(x)$

Answer 1

Hint: According to given in the question we have to determine the ${f^{ - 1}}(x)$ where, $f(x) = \dfrac{{2x - 1}}{{x - 1}}$. So, first of all as mentioned in the question that $f(x) = \dfrac{{2x - 1}}{{x - 1}}$ hence, we have to consider ${f^{ - 1}}(x)$ as $y = \dfrac{{2x - 1}}{{x - 1}}$.
Now, we have to solve the expression as obtained by applying the cross-multiplication and multiplying all the terms obtained after multiplication.

Complete step-by-step answer:
Step 1: First of all as mentioned in the question that $f(x) = \dfrac{{2x - 1}}{{x - 1}}$hence, we have to consider ${f^{ - 1}}(x)$ as,
$ \Rightarrow $$y = \dfrac{{2x - 1}}{{x - 1}}$…………….(1)
Step 2: Now, we have to solve the expression (1) as obtained in the solution step 1 by applying the cross-multiplication and multiplying all the terms obtained after multiplication.
$
   \Rightarrow y(x - 1) = 2x - 1 \\
   \Rightarrow yx - y = 2x - 1.................(2) \\
 $
Step 3: Now, we have to subtract 2x from the both of the sides of the obtained expression (2) as in the solution step 2. Hence,
$
   \Rightarrow yx - y - 2x = 2x - 2x - 1 \\
   \Rightarrow (yx - 2x) - y = - 1.............(3) \\
 $
Step 4: Now, we have to add y in the both sides of the expression (3) as obtained in the solution step 3. Hence,
$
   \Rightarrow (yx - 2x) - y + y = - 1 + y \\
   \Rightarrow yx - 2x = y - 1 \\
 $
Step 5: Now, we have to take y as a common term from the left side of expression as obtained in the solution step 4 above,
\[ \Rightarrow x(y - 2) = y - 1 \\
   \Rightarrow x = \dfrac{{y - 1}}{{y - 2}} \\
 \]

Hence, we have determine the value of ${f^{ - 1}}(x)$where, $f(x) = \dfrac{{2x - 1}}{{x - 1}}$which is \[x = \dfrac{{y - 1}}{{y - 2}}\].

Note:
It is necessary that we have to take ${f^{ - 1}}(x)$ as y to obtain the value of y with the help of the new form of the expression which is $y = \dfrac{{2x - 1}}{{x - 1}}$.
To obtain the value of y we have to subtract 2x in the both sides of the expression as obtained after cross-multiplication and then same as we have to subtract y in the both sides of the expression obtained.