What is the proper way to find the inverse of a function?

Question

I am a little confused on the subject of inverse functions and the methods used to do the transformation from function to inverse. How do you make an inverse? Just so i can avoid any ambiguity in my question, let's change it to the following: Would anyone on this fantastic website be kind enough to list the steps of "inversing", may be the word, the simple function of $$f(x)= \frac{(x-3)}{2}$$

Answer 1

First of all, to type 3/over 2 in latex you should enter "\frac{3}{2}". For more informations you could just search Latex and there will be a lot of supporting instructions you can look at.

Next, once you have a function, say $y=f(x)$ and you wish to find the inverse function of it. The simplest idea is you treat $y$ as a given number and you find the number of $x$ in terms of $y$. Hence you will have a function such that $x=g(y)$ where $g(y)$ is the inverse function you are looking for. Of course, there are many situations that you can not explicitly solve $x$ in terms of $y$ but I think you won't meet those cases in your homework, exams, etc.

Of couse, $f(x) = 2x+3$ is the correct answer.

Answer 2

If a function has an inverse then the roles of $x$ and $y$ are switched. For the function you gave $f(x)=\frac{x-3}{2}$ follow the steps.

1) Replace $f(x)$ by $y$: $$y=\frac{x-3}{2}$$

2) Switch the variables $x$ and $y$: $$x=\frac{y-3}{2}$$

3) Solve for $y$: $$2x=y-3 \Longrightarrow y=2x+3$$

4) Replace $y$ by $f^{-1}$: $$f^{-1}(x)=2x+3$$

Answer 3

I know there's already an accepted answer, but I'm not a fan of how its laid out, and it also has some errors in the algebra. So, here's my version.

Please comment with any questions you may have.

Let $f$ denote the unique function $\mathbb{R} \rightarrow \mathbb{R}$ defined as follows. $$f(x)=\frac{x−3}{2}$$

Problem. Find the inverse of $f$.

Solution. Let $x,y \in \mathbb{R}$ be fixed but arbitrary. Then TFAE.

1. $f(x)=y$
2. $\dfrac{x-3}{2}=y$
3. $x-3=2y$
4. $x=2y+3$

Therefore, the inverse of $f$ is the unique function $g : \mathbb{R} \rightarrow \mathbb{R}$ defined as follows.

$$g(y)=2y+3$$