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

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

Explanation

Solution

According to given in the question we have to determine the f−1(x){f^{ - 1}}(x) where, f(x)=2x−1x−1f(x) = \dfrac{{2x - 1}}{{x - 1}}. So, first of all as mentioned in the question that f(x)=2x−1x−1f(x) = \dfrac{{2x - 1}}{{x - 1}} hence, we have to consider f−1(x){f^{ - 1}}(x) as y=2x−1x−1y = \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)=2x−1x−1f(x) = \dfrac{{2x - 1}}{{x - 1}}hence, we have to consider f−1(x){f^{ - 1}}(x) as,
⇒\Rightarrow y=2x−1x−1y = \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.
⇒y(x−1)=2x−1 ⇒yx−y=2x−1.................(2)  \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,
⇒yx−y−2x=2x−2x−1 ⇒(yx−2x)−y=−1.............(3)  \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,
⇒(yx−2x)−y+y=−1+y ⇒yx−2x=y−1  \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,

⇒x=y−1y−2  \Rightarrow x = \dfrac{{y - 1}}{{y - 2}} \\\

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

Note:
It is necessary that we have to take f−1(x){f^{ - 1}}(x) as y to obtain the value of y with the help of the new form of the expression which is y=2x−1x−1y = \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.