Question

How do you find the inverse of $f(x) = 3x + 1$ ?

Answer 1

An expression containing two variable terms is known as a function; that one term whose value doesn’t depend on the value of the other variable is called the independent variable and the other term is called the dependent variable as it changes with the value of the independent variable. In this question, let “y” (dependent variable) be equal to the given function. Now, we can find the inverse of the function by expressing x in terms of y, such that x is now the dependent variable and y is the independent variable.

Complete step by step answer:
We are given that $f(x) = 3x + 1$
Let $f(x) = y$
$\Rightarrow y = 3x + 1$
To find the inverse of the given function, we express x in the terms of y –
$\Rightarrow 3x = y - 1 \\\ \Rightarrow x = \dfrac{{y - 1}}{3} \\\$
We know that $f(x) = y$
$\Rightarrow x = {f^{ - 1}}(y) \\\ \Rightarrow {f^{ - 1}}(y) = \dfrac{{y - 1}}{3} \\\ \Rightarrow {f^{ - 1}}(x) = \dfrac{{x - 1}}{3} \\\$
Hence, the inverse of the function $f(x) = 3x + 1$ is ${f^{ - 1}}(x) = \dfrac{{x - 1}}{3}$ .

Note: The inverse of a function can simply be defined as the reflection of the given function in the line $y = x$ , the other name of the inverse function is anti-function, as it reverses the process done in the original function, that is, if we put some value of the independent variable in the original value and get some value of the dependent variable as the output, then this output will be the input of the inverse function and give the same value that was used as the independent variable. Not all the functions have an inverse, for the inverse of a function to exist, the function must be a one-one and onto function.