Composite Functions

Summary

This lesson delves into composite functions, contrasting them with function multiplication. It uses f(x) = 3x - 4 and g(x) = x^2 - 3 to demonstrate how to calculate f(g(x)) and g(f(x)). The process involves substituting one function into another, showcasing the steps to find f(g(x)) = 3(x^2 - 3) - 4 and g(f(x)) = (3x - 4)^2 - 3. The lesson further illustrates the evaluation of composite functions with examples, such as f(g(2)) and g(f(-1)), enhancing understanding of function composition.

Takeaways

• 🔢 Composite functions are different from function multiplication; they involve one function 'inside' another.
• 📐 The notation for composite functions is an open circle (e.g., f(g(x))) indicating that g(x) is substituted into f(x).
• 🔄 To find f(g(x)), substitute g(x) into f(x) wherever there is an x in the function f(x).
• 📘 The process of finding g(f(x)) involves substituting f(x) into g(x) wherever there is an x in the function g(x).
• 🧮 Example calculation: f(x) = 3x - 4 and g(x) = x^2 - 3 leads to f(g(x)) = 3(x^2 - 3) - 4 which simplifies to 3x^2 - 13.
• 📊 For g(f(x)), the example given is f(x) = 3x - 4 and g(x) = x^2 - 3, resulting in g(f(x)) = (3x - 4)^2 - 3 which simplifies to 9x^2 - 24x + 13.
• 📈 When evaluating composite functions at specific values, first calculate the inner function's value and then substitute it into the outer function.
• 🔑 The example for f(g(2)) with f(x) = 5x + 2 and g(x) = x^3 - 4 results in f(g(2)) = 22 after finding g(2) = 4.
• 💡 For g(f(-1)), with the same functions, it results in g(f(-1)) = -31 after finding f(-1) = -3 and substituting into g(x).
• 📚 The lesson emphasizes the importance of understanding the order of operations and the correct substitution of values in composite functions.

Q & A

What is the difference between f(x) * g(x) and f(g(x))?

f(x) * g(x) represents the pointwise multiplication of two functions, whereas f(g(x)) is a composite function where g(x) is substituted into f(x).

What is the expression for f(g(x)) if f(x) = 3x - 4 and g(x) = x^2 - 3?

f(g(x)) is calculated by substituting g(x) into f(x), resulting in f(g(x)) = 3(x^2 - 3) - 4, which simplifies to 3x^2 - 9 - 4, or 3x^2 - 13.

How do you find the value of g(f(x)) when f(x) = 3x - 4 and g(x) = x^2 - 3?

To find g(f(x)), you substitute f(x) into g(x), which gives g(f(x)) = (3x - 4)^2 - 3. After expanding and simplifying, it results in 9x^2 - 24x + 16 - 3, or 9x^2 - 24x + 13.

What is the value of f(g(2)) if f(x) = 5x + 2 and g(x) = x^3 - 4?

First, calculate g(2) which is 2^3 - 4 = 8 - 4 = 4. Then, f(g(2)) is f(4) = 5*4 + 2 = 20 + 2 = 22.

How do you evaluate g(f(-1)) given f(x) = 5x + 2 and g(x) = x^3 - 4?

First, find f(-1) which is 5*(-1) + 2 = -5 + 2 = -3. Then, g(f(-1)) is g(-3) = (-3)^3 - 4 = -27 - 4 = -31.

What is the significance of the order of functions in composite functions?

The order of functions in composite functions is significant as it determines which function's output becomes the input for the other function.

Can you provide an example of how to distribute a constant in a composite function?

Yes, in the script, the constant 3 is distributed over x^2 - 3 in f(g(x)) = 3(x^2 - 3) - 4, resulting in 3x^2 - 9 - 4.

What is the FOIL method mentioned in the script, and how is it used?

The FOIL method is used for multiplying two binomials. It stands for First, Outer, Inner, Last, and is used in the script to multiply (3x - 4)(3x - 4).

How does the script demonstrate the process of evaluating composite functions at specific values?

The script demonstrates evaluating composite functions at specific values by first finding the inner function's value at that point and then using it as the input for the outer function.

What is the final result of g(f(-1)) as explained in the script?

The final result of g(f(-1)) is -31, as calculated by first finding f(-1) = -3 and then substituting it into g(x) to get g(-3) = -27 - 4.

Outlines

00:00

📘 Introduction to Composite Functions

This paragraph introduces the concept of composite functions, distinguishing them from simple multiplication of functions. The functions f(x) = 3x - 4 and g(x) = x^2 - 3 are defined, and the process of finding f(g(x)) is explained. It involves substituting g(x) into f(x), which results in the expression 3(x^2 - 3) - 4, simplified to 3x^2 - 13. The paragraph also explains how to find g(f(x)) by substituting f(x) into g(x), leading to the expression (3x - 4)^2 - 3, which simplifies to 9x^2 - 24x + 13 after applying the FOIL method.

Keywords

💡Composite Functions

Composite functions are functions that are formed by applying one function to the result of another. In the video, the concept is introduced by considering two functions, f(x) and g(x), and then defining f(g(x)) and g(f(x)). This is central to the video's theme as it demonstrates how to find the output of one function when the input is the result of another function. For example, f(g(x)) is calculated by substituting g(x) into f(x), which is shown when the script replaces x with g(x) = x^2 - 3 in the function f(x) = 3x - 4.

💡Function Composition

Function composition, often denoted by an open circle (∘), refers to the process of combining two functions to produce a new function. The video script explains this by contrasting it with function multiplication, which uses a closed circle (⋅). The script uses function composition to derive new expressions, such as f(g(x)) = 3(x^2 - 3) - 4, illustrating how one function's output becomes the input for another.

💡Distributive Property

The distributive property is a fundamental algebraic rule that allows us to multiply a term by each term within a parenthesis. In the video, this property is applied when the script calculates f(g(x)) by distributing the 3 across the terms in the expression x^2 - 3, resulting in 3x^2 - 9.

💡FOIL Method

FOIL is a mnemonic for multiplying two binomials, standing for First, Outer, Inner, Last. The video uses FOIL when calculating g(f(x)) by multiplying (3x - 4)(3x - 4), which is expanded to 9x^2 - 12x - 12x + 16. This method is crucial for understanding how to expand expressions when dealing with composite functions.

💡Function Evaluation

Function evaluation involves finding the value of a function for a given input. The video demonstrates this by evaluating f(g(2)) and g(f(-1)). For instance, to find f(g(2)), the script first calculates g(2) = 2^3 - 4, then uses this result as the input for f(x), showing how to plug values into functions to get specific outputs.

💡Exponents

Exponents are used to denote repeated multiplication of a number by itself. In the video, exponents are used when calculating g(x) = x^3 - 4, where x is raised to the power of three. The script shows how to apply exponents when evaluating g(2), which involves calculating 2 cubed.

💡Substitution

Substitution is a technique where a value or expression is replaced with another in a mathematical formula. The video script uses substitution extensively, such as when finding f(g(x)) by replacing x with g(x) in the function f(x). This method is essential for understanding how to work with composite functions.

💡Algebraic Manipulation

Algebraic manipulation refers to the process of transforming algebraic expressions using various mathematical rules and properties. The video demonstrates this by simplifying expressions like 3x^2 - 9 - 4 to 3x^2 - 13, showcasing the steps involved in simplifying expressions after applying operations like distribution.

💡Negative Numbers

Negative numbers are numbers that are less than zero. The video script includes examples involving negative numbers, such as when evaluating f(-1) = 5(-1) + 2, which results in -3. This illustrates how to handle negative values in function evaluation.

💡Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. The video script involves polynomials when calculating expressions like 3x^2 - 13, which is a quadratic polynomial, and demonstrates how to work with these types of expressions in the context of composite functions.

Highlights

Introduction to composite functions where one function is inside another.

Definition of function f(x) as 3x - 4.

Definition of function g(x) as x squared - 3.

Explanation of the difference between composite functions and function multiplication.

Finding f(g(x)) by substituting g(x) into f(x).

Distributing the 3 in the expression 3(x^2 - 3) - 4 to get 3x^2 - 13.

Finding g(f(x)) by substituting f(x) into g(x).

Calculating (3x - 4)^2 to find the value of g(f(x)).

Applying the FOIL method to expand (3x - 4)^2.

Final expression for g(f(x)) is 9x^2 - 24x + 13.

Evaluating f(g(x)) at a specific point, f(g(2)) = 22.

Evaluating g(f(x)) at a specific point, g(f(-1)) = -31.

Procedure for evaluating composite functions step by step.

Importance of correctly substituting values when evaluating composite functions.

Use of algebraic manipulation to simplify expressions in composite functions.

Practical example of evaluating composite functions with real numbers.

Emphasis on the sequential nature of composite functions evaluation.

Transcripts

now in this lesson we're going to talk

about composite functions

so let's say that f of x

is equal to 3x minus 4

and that g of x

let's say it's equal to x squared

minus three

what is f of g

notice that this is different from f

times g

if you see a closed circle

it's multiplication it's 3x minus 4

times x squared minus 3.

but if you see like an open circle

what it means is

it's a composite function

one function is inside of another

g

is inside of f so this is equivalent to

f

of g of x

so how can we find f of g of x

well let's replace g

g

is x squared minus three

and notice that x squared minus three is

inside of f

so we're going to have to replace x

with x squared minus 3. so this is going

to be 3

times x squared minus 3

minus 4.

so now let's distribute the 3.

so that's 3x squared minus 9.

minus four

which is

three x squared minus thirteen

now what about g of f

how can we find the value

of g of f

so this time

f

is inside of g

so you can write it like this

that's an equivalent expression let's

replace f of x with three x minus four

and now

let's replace uh x with three x minus

four

so we're going to take this

and plug it into here

so instead of x squared minus 3 it's 3x

minus 4 squared minus 3.

3x minus 4 squared is 3x minus 4

times 2x minus 4.

so if we foil it

3x times 3x that's going to be

9x squared and then 3x times negative 4

that's

negative 12x

negative 4 times 3x is also negative 12x

and then negative 4 times negative 4 is

16.

negative 12 minus 12 is negative 24

16 minus 3 is 13.

so

this is the value

of

g of f of x

now let's say that f of x

let's say it's 5x plus 2.

and now let's say that g of x

is equal to x cubed

minus four

evaluate this function f

of g

of two

how can we do so

so first let's find the value of

g of two

so let's replace x with two

this is going to be two raised to the

third power minus four

two times two times two three times is

eight

eight minus four is four so g of two

is four so f of g of two is equivalent

to f of four

and now we can take four

and plug it into that equation so it's

five times four plus two

five times four is twenty twenty plus

two is twenty two

so that's how you can find the value of

f of g of two

so try this one

evaluate g

of f

let's say

negative one

so first evaluate f of negative one

that's five times negative one plus two

so make sure you're using this equation

that's negative five plus two

which is negative three

so f of negative one we can replace it

with negative three so now we're looking

for g of negative three which we can

plug it into here

so it's negative three to the third

power

minus four

which is negative 27 minus four and so

the final answer

is negative thirty one so that's how you

can evaluate composite functions

you