00:01
Okay, we're given that f of x is an increasing function, and we have an inverse of f of x, which is g of x inverse.
00:13
And we're asked to show that from 0 to a of f of x t x, that's integral from 0 to f of a, from g of x, t x, that's this a times f of a.
00:26
Okay, so let me draw something else.
00:29
So if we know that f of x is an increasing function, so as we plug in positive numbers of x, we should get positive numbers of y, and that should increase.
00:40
So we have something like this.
00:42
It doesn't need to be a straight line, so it can be something like this.
00:48
Okay, and this is from zero to a, this is what we're considering, correct? so if we take the integral from 0 to a of our f of x function, that should just give us something down here.
01:01
But the inverse of f of x, which is g of x, should look something like this.
01:09
So it should be reflected on the y is equal to x axis.
01:14
So if this is symmetric, i should get something like the same thing here.
01:22
Right...