Question

How do you find the inverse of \[f(x) = \dfrac{{2x - 3}}{{x + 4}}\] ?

Answer 1

Hint: Here in this question we have to find the inverse of the function, since the function is of the form fractional number we use the concept of simple arithmetic operations and on further simplification we obtain the required solution for the given question. To find the inverse we swap the terms.

Complete step by step solution:
In mathematics, an inverse function is a function that "reverses" another function: if the function \[f\] applied to an input x gives a result of y, then applying its inverse function \[g\] to y gives the result x, i.e., \[g(y) = x\] if and only if \[f(x) = y\]. The inverse function of \[f\] is also denoted as \[{f^{ - 1}}\]. Now consider the given function \[f(x) = \dfrac{{2x - 3}}{{x + 4}}\]. As we know that \[f(x) = y\], on substituting it we have,
\[y = \dfrac{{2x - 3}}{{x + 4}}\]
Now swap the variables that is y to x and x to y we have
\[x = \dfrac{{2y - 3}}{{y + 4}}\]
Take (y+4) to LHS we get
\[x(y + 4) = 2y - 3\]

On simplification
\[ xy + 4x = 2y - 3\]
Take \[xy\] to RHS and 3 to LHS we get
\[4x + 3 = 2y - xy\]
Take y as a common in RHS and the above function is written as
\[4x + 3 = y(2 - x)\]
Divide the above equation by (2-x)
\[\dfrac{{4x + 3}}{{2 - x}} = y\]
Therefore it can be written as
\[y = \dfrac{{4x + 3}}{{2 - x}}\]

We can verify by considering the example.Consider \[f(x) = \dfrac{{2x - 3}}{{x + 4}}\], now take x as 1. the value is \[f(2) = \dfrac{{2(2) - 3}}{{2 + 4}} = \dfrac{{4 - 3}}{{2 + 4}} = \dfrac{1}{6}\].
Now consider \[{f^{ - 1}}(x) = \dfrac{{4x + 3}}{{2 - x}}\], now take x as \[\dfrac{1}{6}\] then the value is,
\[{f^{ - 1}}\left( {\dfrac{1}{6}} \right) = \dfrac{{4\left( {\dfrac{1}{6}} \right) + 3}}{{2 - \dfrac{1}{6}}}\\
\Rightarrow{f^{ - 1}}\left( {\dfrac{1}{6}} \right)= \dfrac{{\dfrac{4}{6} + 3}}{{2 - \dfrac{1}{6}}} \\
\Rightarrow{f^{ - 1}}\left( {\dfrac{1}{6}} \right)= \dfrac{{\dfrac{{4 + 18}}{6}}}{{\dfrac{{12 - 1}}{6}}} \\
\Rightarrow{f^{ - 1}}\left( {\dfrac{1}{6}} \right)= \dfrac{{22}}{{11}} \\
\therefore{f^{ - 1}}\left( {\dfrac{1}{6}} \right)= 2\]
Hence verified.

Therefore the inverse of \[f(x) = \dfrac{{2x - 3}}{{x + 4}}\] is \[{f^{ - 1}}(x) = \dfrac{{4x + 3}}{{2 - x}}\].

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 is a fraction on simplification we must know about the concept of LCM. While shifting the terms we must take care of signs.