In Exercises 17 to 26, use composition of functions to determine whether f and g are inverses of

In Exercises 17 to 26, use composition of functions to...

Key Concepts

Inverse Functions

An inverse function essentially reverses the action of the original function. For two functions to be inverses, the composition of one with the other (in both orders) must yield the identity function, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains. This property ensures that applying one function and then its inverse returns you back to the original input.

Composition of Functions

Function composition is the process of applying one function to the results of another. In the context of checking for inverses, you compose the two functions in both orders. If the resulting compositions simplify to the identity function, it confirms that each function undoes the effect of the other, thereby establishing an inverse relationship.

Identity Function

The identity function is a function that always returns its input, typically written as I(x) = x. When verifying inverse functions, both compositions f(g(x)) and g(f(x)) should equal the identity function, indicating that the functions exactly undo each other’s operations.

Transcript

00:01 Show you how to confirm that two equations are in verses of each other.

00:05 So the first thing we want to talk about is what are we doing? so in order to confirm invoices, inverse.

00:12 Excuse me.

00:12 Using two different functions.

00:15 We are looking for the composite functions.

00:18 So that means f of g of x.

00:21 Or you might see it like this f of g of x equals the letter x.

00:28 Okay, so what we're gonna do is we're going to plug the g of x function.

00:34 So this function right here into the f function.

00:39 So what that means is i'm going to take this exact entire function and plug it in for x right here.

00:45 So i'm gonna first rewrite f of g of x is what i'm doing.

00:50 And then instead of the letter x for the f of x function, so notice instead of x cubed plus five, i'm going to replace it with the g of x function.

01:01 So the cubed root of x minus five.

01:05 So where i see x in the fx function, i am replacing it with g.

01:10 Of x is function now everywhere.

01:12 I see an x.

01:13 I would plug in this g fx function.

01:16 So if it was like x cubed plus two x squared plus x plus five, i would plug it in everywhere that i see an x.

01:23 So i'm replacing the x.

01:25 For the ffx function with the g fx function everywhere.

01:29 I see an x now notice i'm taking this function, plugging it in here.

01:34 I'm not plugging it into itself because then you'll go around in circles so you're going to take one function and plug it into the other.

01:40 Now the great thing is it doesn't matter which way you do this because if it works one way it should work the other way.

01:46 So if you plug it in for f.

01:47 Of x and you're like oh that looks really ugly, i'm not sure what's going on...

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