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

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.