Question

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

Answer 1

Hint: 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.