Question

If \[f\left( x \right)\] is a linear function and the slope of \[y = f\left( x \right)\] is \[\dfrac{1}{2}\]​, find the slope of \[y = {f^{ - 1}}\left( x \right)\]

Answer 1

Hint: An algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable is one then the algebraic equation is known as linear equation.
Here we are provided that \[f\left( x \right)\] is a linear function and slope of \[f\left( x \right)\] is \[\dfrac{1}{2}\].
With the slope given we are going to find the value of \[f\left( x \right)\] and then we will find the inverse of the function and then we will find the slope of the inverse function.

Complete step-by-step answer:
It is given that, \[y = f\left( x \right)\] and its slope is \[\dfrac{1}{2}\]
Since the given function is a linear function we know that the equation formed is of the form \[y = mx\] where m is the slope of the equation.
Using the above fact we can write that \[y = \dfrac{1}{2}x\].
We have found that \[y = \dfrac{1}{2}x\] to find inverse of the function
Finding the inverse function we have to take given the function $f(x)$. We want to find the inverse function ${f^{ - 1}}(x)$.
First, replace $f(x)$ with $y$. This is done to make the rest of the process easier.
Replace every $x$ with $y$ and replace every $y$ with an $x$.
Solve the equation from above step for $y$. Replace $y$ with ${f^{ - 1}}(x)$.
\[ \Rightarrow f(x) = y = \dfrac{1}{2}x\]
We will switch \[x\]and\[y\] by\[y\] and\[x\], which is nothing but the interchange of variables, therefore we get,
\[ \Rightarrow x = \dfrac{1}{2}y\]
Let us solve the above equation to find an equation in \[y\], we get,
\[y = 2x \ldots \ldots \left( 1 \right)\]
Let us mark the equation as (1).
Here we will replace the value of y using the fact that\[y = {f^{ - 1}}\left( x \right)\]
Therefore we get \[{f^{ - 1}}(x) = 2x\]
From the initial case we consider \[y = mx\] and compare it with the equation (1). We get the slope m.
Hence we have found the slope of\[y = {f^{ - 1}}\left( x \right)\] as \[2\].

Hence the slope of the inverse function is \[2\].

Note: Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and \[y\]-intercepts therefore we use the formula \[y = mx\]. Solving the equation to find the step for $y$. This is the step where mistakes are most often made so be careful with this step.