Question 1054387
<pre>
{{{"f(x)"}}}{{{""=""}}}{{{sqrt(x)-5}}}

Draw the graph and also the graph of the identity line
which has equation y = x.  The green dotted line is the 
identity line.  The graph of the inverse function f<sup>-1</sup> 
is the reflection of the red graph across the identity line.

{{{drawing(300,300,-7,7,-7,7, graph(300,300,-7,7,-7,7,sqrt(x)-5,x*sqrt(sin(10x))/sqrt(sin(10x)))))}}}

Then substitute y for f(x)

{{{y}}}{{{""=""}}}{{{sqrt(x)-5}}}

Interchange x and y:

{{{x}}}{{{""=""}}}{{{sqrt(y)-5}}}

Solve for y:

{{{x+5}}}{{{""=""}}}{{{sqrt(y)}}}

{{{(x+5)^2}}}{{{""=""}}}{{{y}}}

{{{y}}}{{{""=""}}}{{{(x+5)^2}}}

Draw the graph on the same axes to see if we 
need to restrict it, and if so how:

{{{drawing(300,300,-7,7,-7,7, 
graph(300,300,-7,7,-7,7,(x+5)^2),
graph(300,300,-7,7,-7,7,sqrt(x)-5,x*sqrt(sin(10x))/sqrt(sin(10x)))))}}}
 
So we see that we only want to keep the part of that
graph to the right of x = -5, so it will be the reflection
of the first graph in the green identity line.

So to do that we must restrict the graph on the left 
to value of x greater than or equal to -5, so the graph
on the left will look like this, and contain only the
reflection of the first graph as we see below:

{{{drawing(300,300,-7,7,-7,7, 
graph(300,300,-7,7,-7,7,(x+5)^2*sqrt(x+5)/sqrt(x+5)),
graph(300,300,-7,7,-7,7,sqrt(x)-5,x*sqrt(sin(10x))/sqrt(sin(10x)))))}}}

So we take the equation

{{{y}}}{{{""=""}}}{{{(x+5)^2}}}

Replace y by f<sup>-1</sup>(x)

And put the restriction x &#8805; -5 after it:

Answer: f<sup>-1</sup>(x) = (x+5)<sup>2</sup>; x &#8805; -5

You don't have that listed with the correct restriction,

Also restrictions are usually always give with x rather than 
y.  All the choices you have listed above have only y, not x.
Did you copy the problem wrong?

Edwin</pre>