Function Inversion

Question

I have this equation:

$f(x) = x + 3$

I need to invert it. Could somebody do it for me and explain how it's done?

Thank you!

Answer 1

If we suppose that $y=f(x)$, then from $y=x+3$ we want to re-arrange the equation to give $x$ as a function of $y$, which we can do quite simply by subtracting $3$ from both sides to give $x=y-3$. Then as $y=f(x)$, we can take the inverse of this to get $x=f^{-1}(y)$, and thus the inverse function is $f^{-1}(y)=y-3$, or as now $y$ is arbitary, $f^{-1}(x)=x-3$.

Answer 2

By definition, the inverse of a function $f$ is the function $g$ such that $f \circ g=\mathrm{id}$ where $\mathrm{id}$ denotes the identity function, i.e., $\mathrm{id}(x)=x$ for all $x$. So, we're seeking a function which satisfies $$g(f(x))=x$$ for all $x$ or equivalently $$g(x+3)=x$$ for all $x$. With a change of variables $u=x+3$, we see we want $$g(u)=u-3$$ for all $u$.

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The above can be reformulated into a familiar form: if $y=f(x)$ and $$y=x+3$$ then $f^{-1}$ can be found by swapping the $x$ and $y$ in the equation , i.e., $$x=y+3,$$ then solving for $y$.

Answer 3

It is this:

$f^{-1}(x)=x-3$.

It can be done as follows: $x=f(x)-3$. Let $f^{-1}(x)$ take place of $x$ and $x$ take place of $f(x)$.

Answer 4

If the function is a mapping from $\mathbb{R}$ to $\mathbb{R}$, $x\in\mathbb{R}$ to $y\in\mathbb{R}$, $(x\rightarrow y)$, the inverse function is a mapping from $y$ to $x$. In this case, you have: $f(x)=x+3$, so: $x=f(x)-3$. This is a simple case because there is only one value of $x$ giving a value for $f(x)$. Not every function has this property. For example $f(x)=\sin(x)$ has the inverse function: $x=\arcsin\left(f(x)\right)$. There are infinite values of $f(x)$ giving the same $x$.

Answer 5

There are three main steps to find the inverse:

First write your function using y in place of $f(x)$, so you have $y = x + 3$.

The next step is to swap the $x$ and $y$ variables everywhere they appear in your equation. When you do this, you get $x = y - 3$.

The final step is to solve for y again so that you just have y= on one side of the equation. You can do this by subtracting 3 from both sides, and you get $y = x - 3$. Since this equation is also a function (meaning for each $x$ value you put in, there's only one corresponding $y$ value), this is your inverse function. You can use function notation but with the negative 1 superscript, i.e. $f^{-1}(x) = x - 3$.

Good luck! If you need more help on finding the inverse of a function, this video explains it and gives examples.