# Looking for a simple interpretation

$$f(x) = 10-x$$

If I plugin $$x=2$$, I get $$f(2)=10-2=8$$.
If I want to know what I must plugin to get $$8$$, again I simply plugin $$8$$ into $$f(x)$$ : $$f(8) = 10-8=2$$.
One can conclude $$f(x) = f^{-1}(x)$$.

But this is not so clear to me. Specifically for this particular function $$f(x)=10-x$$, I'm trying to visualize using examples like, taking away $$2$$ apples from $$10$$ etc. But they don't seem to make much sense. I'm wondering if you have any other means to interpret $$f(x)$$.. Thanks in advance :)

A function is self-inverse (that is $$f(x)=f^{-1}(x)$$) , if we have $$f(f(x))=x$$ This can be easily verified for this function.

• Another way to find this equation is to calculate $f^{-1}(x)$ for $f(x)=10-x$. We have $y=10-x$, which we can transform into $x=10-y$. Exchanging $x$ and $y$ leads to $y=10-x$, the original function. – Peter Dec 15 '18 at 10:14
• Finally, you can verify the equation graphically by noting that the line $f(x)=10-x$ is symmetric with respect to the line $f(x)=x$, hence it is self-inverse. – Peter Dec 15 '18 at 10:18
• Ahh right, any graph symmetric about $y=x$ is its own inverse because the points $(x,y)$ and $(y, x)$ exist on the same graph. Thank you so much :) – rsadhvika Dec 15 '18 at 10:20
• Just for the record, a function that equals its own inverse is called an involution. – Andreas Rejbrand Dec 15 '18 at 13:59

The graph of the function $$y=10-x$$ is a straight line with the slope $$-1$$ and $$y$$-intercept $$10$$. If you reflect (see "Graph of the inverse" section) the straight line across $$y=x$$, you will get the line itself, because they are perpendicular. Hence the function is self-inverse. See the graph:

$$\hspace{3cm}$$

$$y=10-x \iff x=10-y$$. This shows that $$f(x) = f^{-1}(x)$$.

Using your analogy, this function partitions the apples into a "taken away group" and a "left over group".

If you take away 2, you have 8 left over.
If you have 8 left over, you took away 2.

This function is a map between # taken away <-> # left over.