I have this homework problem that I'm having trouble with. It is:

If f(x) = 1 - x[sup]3[/sup] and f[sup]-1[/sup] is the inverse of f, how many solutions does the equation f(x) = f[sup]-1[/sup] have?

I did the inverse as such:

x = [sup]3[/sup]√(1-x)

It looks to me as if they equal each other twice but that is not a choice. :( The choices are:
A. 0
B. 1
C. 3
D. 5
E. 6

If you graph both equations, you'll find 5 intersections.  Where there's intersections, that shows the two equations are equal with those X values.

Thus, D is the answer.

Edit:  I wrongly corrected you.  Hope it isn't too late to make the correction.  I just realized it today when I was thinking about it again.  The error was just my wrong method of finding a function's inverse. :-\

I graphed it on my TI87 and used ZBox around the intersection. It is D.

So how is your answer wrong?

y = 1 - x[sup]3[/sup]
x = 1 - y[sup]3[/sup]
1 - x = y[sup]3[/sup]
[sup]3[/sup]√(1 - x) = y

[quote author=Shout link=topic=13779.msg140747#msg140747 date=1136519250]
I graphed it on my TI87 and used ZBox around the intersection. It is D.

So how is your answer wrong?

y = 1 - x[sup]3[/sup]
x = 1 - y[sup]3[/sup]
1 - x = y[sup]3[/sup]
[sup]3[/sup]√(1 - x) = y
[/quote]

I had a correction on your answer that was wrong because I didn't use the correct method for finding an inverse.  I just deleted that bit of my post to avoid confusion.

I don't think it can be solved easily without graphing.

Mathematica:
Plot[{1 - x^3, (1 - x)^(1/3), -(x - 1)^(1/3)}, {x, -1, 1.5}]