for the given functions f and g, find: (a) f ∘ g (b) g ∘ f (c) f ∘ f (d) g ∘ g

step 1 All right, we're composing two rational functions together. So our first one is F of G. We are g

For the given functions f and g, find: (a) f ∘ g ...

Key Concepts

Function Composition

Function composition involves applying one function to the results of another. It is denoted as (f ? g)(x) = f(g(x)), meaning that you first evaluate g at x and then apply f to the result. This concept is fundamental in understanding how functions can be combined to create new functions.

Domain of Composite Functions

The domain of a composite function f ? g consists of all values x for which g(x) is defined and f(g(x)) is defined. This requires careful consideration to ensure that the output of the inner function (g) lies within the domain of the outer function (f), which is essential to avoid undefined expressions.

Handling Domain Restrictions in Rational Functions

When dealing with rational functions, domain restrictions primarily arise from the need to avoid division by zero. In composite functions involving rational expressions, it is crucial to identify and exclude those x-values that either make the denominator of the inner function zero or cause the substituted value in the outer function's denominator to become zero. This careful analysis ensures the composite function is well-defined.

Transcript

00:07 All right, we're composing two rational functions together.

00:11 So our first one is f of g.

00:13 We are going to be plugging in g into f, so that's the same thing as saying f of x plus 2 over x minus 3.

00:21 And this is going to be a little crazy here, but let's try it.

00:23 We have x plus 2 over x minus 3 and then subtract 5, all over x plus 2 over x minus 3, and then subtract three.

00:39 Okay.

00:40 So we have a complex fraction, and our goal is to get a fraction over a fraction so that we can keep change flip and get into one single fraction, not with a fraction over a fraction.

00:51 So my first play here is to multiply in the numerator x minus 3 over itself because that'll make the same denominator.

01:02 I do have to make sure when i distribute this negative 5, it goes to both terms.

01:07 So i will have x plus 2 over x minus 3 minus 5x plus 15 over x minus 3.

01:19 And this is great news because now i have, at least in the numerator, a two fractions with the same denominator.

01:27 I'm going to do the same thing down here and luckily it's actually the same thing.

01:31 So i always say as ugly as this thing is, right here that i circled, that's still one.

01:38 So one times three is still going to be three.

01:40 It's just changing the way it looks.

01:45 When i multiply this negative three, you have to be careful that i go to both terms.

01:53 And i'm actually noticing up here, this should be a plus with a negative 5x.

01:57 I'm sorry to backtrack.

01:59 I'm just noticing i'm making that error.

02:00 I don't want to do that down here.

02:02 A negative three times a positive x is going to be negative 3x and a negative 3 times a negative 3 is positive 9.

02:10 All right.

02:13 And all together i can now combine this into one massive fraction over fraction by combining the like terms of the numerator here and the numerator here.

02:24 All right.

02:25 So when i see that i have 1x and negative 5x.

02:30 So i'm going to go over here.

02:31 I'm going to have to readjust everything.

02:33 Negative 4x, 2 and 15 makes 17.

02:37 And then the denominator is actually just x over three and then in the numerator of the denominator i have x and negative 3x so that's negative 2x 2 and 2 and 9 makes 11 and then my denominator here is x minus 3 now what i can do is because i have now a fraction a single fraction over another fraction i can keep change flip so i can keep the top fraction change this division bar which is that that's just what a fraction is division change it to multiplication and then flip or make the reciprocal of the denominator fraction that's really helpful in this case because i have a cancellation here and i'm left with negative 4x plus 17 all over negative 2x plus 11 all right, it's going to be a little bit longer problem, so let's just scroll up.

03:44 My next is g of f.

03:48 All right, and looking in the textbook, i see for this problem, that's going to be g of x plus 5 over x plus 1.

04:02 All right, which is going to be equally complicated for this composition...

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