Question

How do you find the inverse of \[f\left( x \right) = 3 - 2x\] ?

Answer 1

Hint: Here in this question, we have to find the inverse of the given function y or \[f(x)\]. The inverse of a function is denoted by \[{f^{ - 1}}(x)\]. Here first we have to write the function in terms of x and then we have to solve for y using mathematics operations and simplification we get the required solution.

Complete step by step solution:
An inverse function or an anti-function is defined as a function, which can reverse into another function. In simple words, if any function “\[f\]” takes \[x\] to \[y\] then, the inverse of “\[f\]” will take \[y\] to \[x\]. If the function is denoted by ‘\[f\]’ or ‘\[F\]’, then the inverse function is denoted by \[{f^{ - 1}}\] or \[{F^{ - 1}}\]. i.e, If \[f\] and \[g\] are inverse functions, then \[f\left( x \right) = y\] if and only if \[g\left( y \right) = x\]. Consider the given function
\[f\left( x \right) = 3 - 2x\]
\[\Rightarrow y = 3 - 2x\]--------(1)
switch the \[x\]'s and the \[y\]'s means replace \[x\] as \[y\] and \[y\] as \[x\]. i.e., \[f(x)\] is a substitute for "\[y\]".

Equation (1) can be written as function of \[x\]i.e.,
\[x = 3 - 2y\]------(2)
Now, to find the inverse we have to solve the equation (2) for \[y\].
Subtract 3 on both side by, then
\[x - 3 = 3 - 2y - 3\]
On simplification, we get
\[x - 3 = - 2y\]
On rearranging
\[- 2y = x - 3\]
Multiply both side by -1, then
\[2y = 3 - x\]
Divide both side by 2
\[y = \dfrac{{3 - x}}{2}\]
\[ \Rightarrow \,\,\,y = \dfrac{3}{2} - \dfrac{x}{2}\]
\[ \therefore\,\,\,{f^{ - 1}}\left( x \right) = \dfrac{3}{2} - \dfrac{x}{2}\]

Hence, the inverse of a function \[f\left( x \right) = 3 - 2x\] is \[{f^{ - 1}}\left( x \right) = \dfrac{3}{2} - \dfrac{x}{2}\].

Note: We must know about the simple arithmetic operations. To find the inverse we swap the y variable into x and simplify the equation and determine the value for y. Since the given question contains a simple equation on simplification we obtain the result. While shifting the terms we must take care of signs.