Question

How do you find the inverse of $f\left( x \right) = 10x$?

Answer 1

Given a function. We have to find the inverse of the function. First, we will set the function as y. Then, swap the variables x and y. solve the function for y. Then, set the function as the inverse of the function.

Complete step by step solution:
We are given the function, $f\left( x \right) = 10x$

Here, the independent variable is $x$ and the dependent variable is $f\left( x \right)$

Now, we will set the function as y, by assuming $f\left( x \right) = y$

$\Rightarrow y = 10x$

Now, swap the variables x and y in the function.

$\Rightarrow x = 10y$

Now, solve the equation for y, by dividing both sides of the equation by 10.

$\Rightarrow y = \dfrac{x}{{10}}$

Now, replace y by ${f^{ - 1}}\left( x \right)$

$\Rightarrow {f^{ - 1}}\left( x \right) = \dfrac{x}{{10}}$

Hence the inverse of $f\left( x \right) = 10x$ is ${f^{ - 1}}\left( x \right) = \dfrac{x}{{10}}$.

Note: The students please note that the inverse of the function is basically the reciprocal of the given function. If any function f takes x as its input and gives y as output, then the inverse of f, will take y as input and x as output. Students must always remember that the inverse of the function is denoted by ${f^{ - 1}}\left( x \right)$. Also, the operations which are performed in the original function are also inverted, such as if the variable x is multiplied by some constant value in the original function, then in the inverse function, the variable is divided by the same value.
Some properties of function are as follows:
There must exist a symmetry relationship between the original function and the inverse of the function, therefore ${\left( {{f^{ - 1}}} \right)^{ - 1}} = f$
The inverse of the function if it exists, then the properties of the inverse must be unique.