How do I reflect a function about a specific line?

Starting with the graph of $f(x) = 3^x$, write the equation of the graph that results from reflecting $f(x)$ about the line $x=3$.

I thought that it would be $f(x) = 3^{-x-3}$ (aka shift it three units to the right and reflect it), but it's wrong.

The right answer is $f(x) = 3^{-x+6}$ but I just can't get to it!

An explained step by step would be appreciated so I can follow what is being done.

Your idea will work if you just carry it fully through. First shift three units to the left, so the line of reflection becomes the y axis, then flip, and finally remember to shift three units back to the right to put the center line back where it belongs.

(This gives the $f(6-x)$ solution you already know).

• What if I wanted to reflect it on the curve $g \left( x \right) = x$?
– Royi
Nov 8 '15 at 1:46
• Reflecting a function along the line y = x is the same as computing the inverse of the function. So use a method for computing the inverse of a function to find the reflection about the line y = x. Feb 18 '20 at 21:23

Replace $x$ with $6-x$. This works because if $x=3+t$, then $6-x=3-t$.

Or, in words: if $x$ is $t$ units to the right from $3$, then $6-x$ is $t$ units to the left from $3$.

• I see that in essence x is replaced with 6-x, but how would I go about determining that is what needs to be done? Sep 12 '11 at 20:09
• @Jakub, please see my edit. Sep 12 '11 at 20:11