00:01
Let's find the inverse of this function.
00:04
So we'll start by changing f of x to y, and then we're going to switch x and y, and then we're going to multiply both sides of the fraction by y minus 6, both sides of the equation, rather.
00:28
So that's going to look like this, because those y minus 6es will cancel.
00:40
So let's distribute, and i need both ys on one side, so i'm going to subtract y from both sides.
00:54
And i'm going to add 6x to both sides.
00:58
So xy minus y, adding 6x to both sides, equals 6x plus 5.
01:10
Okay, last step, i'm going to factor out the y, which leads me with x minus 1, and i'm going to divide by x minus 1.
01:22
So my final answer, and we're best to put that in function notation, and this is my notation for the inverse.
01:38
Also keep in mind, there is one domain exclusion.
01:42
X cannot equal one in this problem.
01:53
Okay, that was the easy part.
01:55
Now, let's prove that they're inverses by showing the compositions.
01:59
So on this first composition, i'm going to start with the inverse function, but everywhere that i have an x, i'm going to replace that x with the original function.
02:14
So it's going to look like this, six times original function plus five.
02:25
And instead of saying five, i'm actually going to use five like this.
02:33
So that i have a common denominator here.
02:38
Let's do the bottom.
02:40
Instead of x, again, i'm going to write x plus 5 over x minus 6.
02:47
And instead of writing just one, i'm going to call this x minus 6 over x minus 6 so that i have a common denominator.
02:57
Okay, from here, i'm going to write the top as a single fraction with the denominator of x minus 6.
03:04
So if i distribute, i get 6x plus 30 plus 5x minus 30.
03:15
Now, realize this is just a division sign.
03:19
So i'm dividing by, again, i'm going to go to a single fraction, x plus 5, and i'm going to push this negative sign through this x minus 6.
03:30
So it's going to be minus x plus 6...