Find g ∘f and f^∘ g for the given functions f and g . f(x)=|2 x+1|, g(x)=3 x^2-1

Name: Problem 48, Chapter 1: Essentials of Precalculus
Uploaded: 2021-08-20T14:10:16-08:00
Duration: 4 min 11 s
Channel: Curtis Becker
Description: Find g ∘f and f^∘ g for the given functions f and g . f(x)=|2 x+1|, g(x)=3 x^2-1

step 1 Hi there. In this video we're going to look at composite functions. So we have two functions on

Find g ∘f and f^∘ g for the given functions f and g ....

Key Concepts

Function Composition

Function composition is the process of applying one function to the output of another function. It involves substituting the entire expression of one function into the other, as seen in expressions like g(f(x)) or f(g(x)). This concept is fundamental in mathematics because it allows complex functions to be built from simpler components and requires careful attention to the order in which functions are applied.

Absolute Value Function

The absolute value function returns the nonnegative magnitude of a number or expression, regardless of its sign. It is a crucial concept when dealing with functions that include absolute values, as it may lead to piecewise behaviors in the function's definition. Understanding how the absolute value affects the output is key when the function is involved in further operations such as composition.

Quadratic Function

A quadratic function is generally expressed in the form ax² + bx + c and represents a parabola when graphed. In the context of composition, recognizing a quadratic function is important because squaring operations significantly influence the structure and range of the resulting function. This understanding aids in predicting the behavior of composite functions that use quadratics.

Transcript

0:00 Hi there.

00:01 In this video we're going to look at composite functions.

00:06 So we have two functions on the left, f of x and blue, g of x and green.

00:10 And we're going to be looking at what we call a composite function or a composition of two functions.

00:15 If you've ever seen the movie inception, where they have a dream inside a dream, well, it's this compositions of this kind of concept in algebra where we have a function inside of a function.

00:26 So if we look at our notation right here, you're going to see that the function, f is listed.

00:31 The function g is listed, and then this little circle, open circle in the middle.

00:35 All right.

00:37 And this is the symbol for composition.

00:41 So just like an addition, subtraction, multiplication, vision, or square root symbol tell us what kind of math we need to perform.

00:48 The composition symbol is doing something similar.

00:51 It's telling us that we're going to input or substitute the function g into f wherever we see the variable x.

01:02 All right so the second function is going into the first function all right so i'm going to start by just writing function f because that's where g is going to be going into in just a moment so we have two x plus one it's the absolute value of that i just remember this absolute value telling you the distance the number is from zero is what that function communicates to us all right now instead of x here within this function here's where the composition comes in we're going to to replace that entire x with our function g so it's like substitution right substituting that function in you think you do this when you evaluate a function for a number right you replace the variable with that given number in this particular case in compositions we replace the variable with another function or expression all right let's go ahead and let's see if we can't simplify this up a little bit finish our function.

02:08 I'm going to use distribution here to come up with a cleaned up version of this function.

02:21 And i can combine like terms one more time.

02:25 We've created our function, our composite function, g into f.

02:39 This can work the other way as well.

02:42 All right...

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