How do I reflect a function about a specific line?

Question

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.

Thanks in advance!

Rasmus · Answer 1 · 2011-09-12T20:23:03.093

3

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 $3$.

edited Sep 12 '11 at 20:23

answered Sep 12 '11 at 20:04

Rasmus

18,946

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? – kubasub Sep 12 '11 at 20:09
@Jakub, please see my edit. – Rasmus Sep 12 '11 at 20:11

hmakholm left over Monica · Accepted Answer · 2011-09-12T20:35:43.570

3

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).

edited Sep 12 '11 at 20:35

answered Sep 12 '11 at 20:09

hmakholm left over Monica

291,611

2

What if I wanted to reflect it on the curve $ g \left( x \right) = x $? – Royi Nov 08 '15 at 01:46
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. – user10 Feb 18 '20 at 21:23

How do I reflect a function about a specific line?

2 Answers2

Linked