00:04
This question asks, given f of x and g of x are these functions inverses of one another? if you're asked if two functions are inverses, you want to find out if f of g of x is x or if g of f of x is x.
00:23
They should both be equal to x.
00:26
Two functions are inverses.
00:28
If i plug in x to the first one, take its result, plug it into g, and it brings me back to the original starting point.
00:38
That's the big idea.
00:39
So f of g of x, given these two, is f of one -fourth of x.
00:46
That's g of x.
00:48
F of one -fourth of x is four times one -fourth of x, which is x.
00:55
So yes, those are inverses of one another.
00:58
Letter b, you're given that f of x is 3x plus one, and g of x is 3x minus 1.
01:08
They seem like opposites, but i'm feeling like this one's going to be no.
01:14
To decide, i find f of g of x, which means take f of x, and in place of the x, i put the function for g.
01:26
So that's 3 times 3x minus 1 plus 1.
01:32
That's 9x minus 3 plus 1, which is not x.
01:40
So this one is no, they're not inverses.
01:44
Letter c, f is the cube root of x minus 2, and g is x cubed plus 2.
01:58
Now some students try checking for inverses by plugging in a single value.
02:04
Just because a single value works in the inverse doesn't mean that the whole function is in our inverses of one another.
02:14
And i don't prove that they're inverses by finding the inverse of one and showing it equals the other.
02:21
When they want you to verify the inverses, they want you to use this definition up here.
02:26
So that's what we are doing.
02:29
F of g of x is f of x cubed plus 2, which means i take the f function and in place of the x again, i'll put the function g of x.
02:43
So that's in place of this x.
02:48
I put x cubed plus 2.
02:52
X cubed plus 2 minus 2 is x cubed.
02:59
And i get x.
03:00
So yes, those are inverses.
03:04
Letter d is a little more straightforward.
03:08
F is x fourth.
03:10
G is the fourth root of x.
03:14
So f of g of x...