Question

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

Answer 1

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.