Question

Find the inverse of $f(x) = 3x - 6$ and is it a function?

Answer 1

We write the function in terms of $x$ to function in terms of $y$. Switch the variables from $x$ to $y$ and from $y$ to $x$. Use arithmetic operations and shift values from one side to another and write the new function as the inverse of the function.
Inverse of a function $y = f(x)$ is the function that undoes the action of the function. A function $g$ is the inverse of the function $f$ if whenever $y = f(x)$ then $x = g(y)$.

Complete step by step solution:
We are given the function $f(x) = 3x - 6$
This is a function in which the independent variable is $x$ and the dependent variable is $f(x)$.
Let us assume $f(x) = y$
Then we can write $y = 3x - 6$.
Now we switch the variables from $x$ to $y$ and vice versa in the function
$\Rightarrow x = 3y - 6$
Shift 6 to left hand side of the equation
$\Rightarrow x + 6 = 3y$
Divide both sides of the equation by 3
$\Rightarrow \dfrac{{x + 6}}{3} = \dfrac{{3y}}{3}$
Cancel same factors from numerator and denominator and both sides of the equation
$\Rightarrow y = \dfrac{x}{3} + 2$
Now again interchange the variables in the equation
$\Rightarrow x = \dfrac{y}{3} + 2$
Substitute the value of $y = f(x)$ then the inverse of $f(x)$will be ${f^{ - 1}}(x)$.
$\Rightarrow {f^{ - 1}}(x) = \dfrac{x}{3} + 2$
Since ${f^{ - 1}}(x)$ can be expressed as a function of $x$, then the inverse of the function exists.
So, the inverse of the function $f(x) = 3x - 6$ is ${f^{ - 1}}(x) = \dfrac{x}{3} + 2$
$\therefore$ The inverse of the function $f(x) = 3x - 6$ is ${f^{ - 1}}(x) = \dfrac{x}{3} + 2$ and the inverse function is also a function.

Note: Do not write the inverse as the same function just by interchanging the variables with each other. Keep in mind the inverse function will be a function of the opposite variable that of the original function variable.