Question 1178867
<pre><b>
She's right except she didn't tell you that one of those, H(x) is ITS OWN
inverse!  Yes, that's right!  A function can be its own inverse.

She showed you how to find inverses by four steps, although she only
mentioned step 2, and assumed you knew the other 3 steps.

1. Replace F(x), G(x), or H(x), by y, respectively.
2. Interchange x and y.
3. Solve for y.
4. Replace y by F<sup>-1</sup>(x), G<sup>-1</sup>(x), or H<sup>-1</sup>(x), respectively.  

Let's follow those 4 steps with H(x)=9/x

    H(x)=9/x
1.     y=9/x
2.     x=9/y
3.    xy=9
       y=9/x
4. H<sup>-1</sup>=9/x  

How about that! H(x) and H<sup>-1</sup>(x)=9/x both equal 9/x, so
that means H(x) is ITS OWN inverse!

She also didn't show you what the graph of a function and its inverse look
like graphically.

I will draw F(x)=9x in blue, G(x)=x/9 in green, on the same set of axes and
also a dotted graph in red of y=x, which is a line that goes 45° through the
origin. It's often called the "IDENTIty line" because its equation, y=x,
shows that it is the case where y and x are IDENTIcal.

{{{drawing(400,400,-10,10,-10,10, 
green(locate(7,2,G(x))), blue(line(-2,-18,2,18),locate(1,7,F(x))),
graph(400,400,-10,10,-10,10,x*sqrt(sin(7x))/sqrt(sin(7x))),
red(locate(7,7,y=x)),
green(line(-18,-2,18,2))),   )}}}

Notice that F(x) and G(x), which are INVERSES are REFLECTIONS of EACH OTHER
in (or across) the dotted IDENTIty line.

Any function's graph and its inverse's graph are always REFLECTIONS of EACH
OTHER in (or across) the IDENTIty line. 

Now let's draw the graph of H(x) in green, and also the identity line y=x:

{{{drawing(400,400,-10,10,-10,10, 

graph(400,400,-10,10,-10,10,11,9/x),
graph(400,400,-10,10,-10,10,x*sqrt(sin(7x))/sqrt(sin(7x))),

red(locate(7,7,y=x))   )}}}

See how the graph is ITS OWN REFLECTION in (or across) the dotted
IDENTITY LINE? H(x) is ITS OWN INVERSE!

Edwin</pre></b>