Function Notation with an Equation

Summary

This video script delves into the application of function notation through equations, providing a clear explanation of how to interpret and calculate outputs for given inputs. It emphasizes the importance of understanding the function's formula and order of operations. The script walks through several examples, including handling absolute values and constants, to demonstrate how to apply function notation in algebraic expressions. It aims to clarify common mistakes and misconceptions, encouraging viewers to practice and become proficient in using function notation.

Takeaways

• 📚 Function notation is a common mathematical tool used to represent the relationship between inputs and outputs in equations.
• 🔍 The function notation typically includes a function name (like F, G, or H) followed by the input variable (X) and an equals sign, indicating the output (Y).
• 📐 The formula for a function provides a generic way to calculate outputs for any given input, unlike visual representations which show specific mappings.
• 🔑 The variable part of the equation is the input (X), while the function name (like 'f') remains constant and represents the operation being performed.
• 🔄 When using a function formula, you substitute the variable (X) with the desired input value to calculate the corresponding output.
• 🧩 Understanding the order of operations is crucial when substituting inputs into a function formula to ensure correct calculations.
• 📘 The script provides examples of how to calculate outputs for given inputs using function formulas, emphasizing the importance of following the correct mathematical procedures.
• 📉 The concept of absolute value is introduced in the script, which is denoted by vertical bars and requires special handling when substituted into a function.
• 🔢 Functions can be simple or complex, and the script demonstrates how to handle both, including polynomials and functions with absolute values.
• 📝 The script also covers how to input expressions (like X + 1) into functions, resulting in outputs that may involve variables rather than specific numbers.
• 🔄 The process of substituting inputs into functions and simplifying expressions is a fundamental skill in algebra, which the script aims to reinforce.

Q & A

What is the basic concept of function notation?

Function notation is a way to represent a function where you write the function name, followed by the input variable in parentheses, and then an equals sign and the expression that defines the output.

How is a function defined using an equation?

A function is defined using an equation by stating the function name, followed by the input variable, an equals sign, and a formula that calculates the output for any given input.

What does 'f(x) = 2(x^2 - 3) / (x - 4)' represent in function notation?

This represents a function named 'f' that takes an input 'x' and outputs the result of the formula '2(x^2 - 3) / (x - 4)'.

What is the significance of the order of operations in calculating the output of a function?

The order of operations is crucial in ensuring that the correct output is calculated. It dictates the sequence in which operations within the function's formula should be performed, such as parentheses, exponents, multiplication/division, and addition/subtraction.

How do you calculate the output of the function f(x) when x is 2?

To calculate the output when x is 2, you replace every instance of 'x' in the function's formula with 2, then follow the order of operations to compute the result, which in this case is -5/2 or negative five-halves.

What does the vertical bar symbol '|' represent in mathematics?

The vertical bar symbol '|' represents the absolute value in mathematics, which means the non-negative value of whatever is inside the bars, regardless of its original sign.

How does the function G(x) handle the absolute value in its formula?

The function G(x) first calculates the value inside the absolute value bars, then takes the absolute value of that result, and finally multiplies it by -2 and adds 4 to get the output.

What is the output of the function H(x) for any input x?

The output of the function H(x) is always -17, as the function is defined to output this value regardless of the input x, because x does not appear in the formula defining H(x).

How can you input an expression like 'x + 1' into a function?

To input an expression like 'x + 1' into a function, you replace every 'x' in the function's formula with 'x + 1' and then simplify the resulting expression to find the output in terms of x.

Why is it important to understand the definition of a function when using function notation?

Understanding the definition of a function is important because it provides the specific formula or rule that determines how the function will transform an input into an output, which is essential for accurate calculations.

Can you provide an example of how to calculate the output of a function with a variable input like 'x + 1'?

Sure, using the function f(x) = 3x^2 - 2x + 1, if the input is 'x + 1', you replace 'x' with 'x + 1' to get f(x + 1) = 3(x + 1)^2 - 2(x + 1) + 1, then simplify to get 3x^2 + 4x + 2 as the output expression.

Outlines

00:00

📚 Introduction to Function Notation with Equations

This paragraph introduces the concept of function notation using equations, as opposed to visual representations like mappings or graphs. It explains the basic structure of function notation, where a function name (e.g., F, G, H) is followed by an input variable (X) and an equals sign, with the output (Y) being calculated based on the input. The paragraph provides an example function, f(X) = (2X^2 - 3) / (X - 4), and emphasizes the importance of understanding the variable part of the equation and the function name as constant. It illustrates how to use the formula to calculate outputs for any given input, highlighting the process of substituting the input value into the formula and performing the calculation.

05:02

🔍 Calculating Outputs with Function Notation

This paragraph delves into the process of calculating specific outputs for a given function using the example of f(2). It explains the importance of understanding the order of operations when substituting values into a function. The paragraph demonstrates the calculation step by step, squaring the input, multiplying by the coefficient, and then performing the necessary subtraction and division to arrive at the output, which in this case is -5/2 or -2.5. The explanation serves to clarify how to handle algebraic calculations within function notation and emphasizes common mistakes to avoid.

10:05

📘 Applying Function Notation to Various Examples

The paragraph presents several functions, f(X), g(X), and h(X), with different algebraic expressions, including polynomials and absolute values. It then provides examples of calculating outputs for these functions with various inputs, such as f(-5), g(-6), and h(4.1). The calculations are shown in detail, with special attention given to the handling of absolute values and the process of substitution. The paragraph also touches on the concept of functions where the output is independent of the input, as demonstrated with h(X), which always outputs -17 regardless of the input value.

📘 Further Examples of Function Application

Continuing from the previous paragraph, this section provides additional examples of applying function notation with variable inputs. It demonstrates how to replace the input variable X with an expression, such as X + 1, in the function definition and then carry out the calculation. The process involves squaring the expression, distributing coefficients, and combining like terms to arrive at an output that may involve the variable X. This paragraph reinforces the understanding of function notation by showing how to handle more complex inputs and emphasizes the importance of algebraic manipulation within function definitions.

Keywords

💡Function Notation

Function notation is a mathematical shorthand used to describe the relationship between an input and an output within a function. In the video, function notation is represented as 'f(x) = y', where 'f' is the function name, 'x' is the input, and 'y' is the output. The script discusses how to use this notation with equations to define functions and calculate outputs, emphasizing the importance of understanding function notation for analyzing mathematical relationships.

💡Equation

An equation in mathematics is a statement that asserts the equality of two expressions. In the context of the video, equations are used to define functions, such as 'f(x) = 2x^2 - 3 / (x - 4)'. The script uses equations to illustrate how to calculate the output of a function for a given input, highlighting the process of substituting values into the equation to find the result.

💡Input

Input refers to the value or variable that is fed into a function to generate an output. The video script uses the term 'input' to describe the value that is placed into the 'F machine', or function, to produce a result. For example, when calculating 'f(2)', the input is the number 2, which is substituted into the function's equation to determine the output.

💡Output

Output is the result produced by a function when an input is processed. The script explains that the output is the final value obtained after substituting the input into the function's equation. For instance, when the input 2 is used in the function 'f(x)', the output is '-5/2', which is calculated by following the function's formula.

💡Variable

A variable in mathematics is a symbol, often a letter, that represents an unknown or changeable value. In the video, 'x' is used as a variable to represent the input to the function. The script emphasizes that variables can take on any value, and the function's formula will calculate the corresponding output for that input.

💡Order of Operations

The order of operations is a set of rules in mathematics that determines the sequence in which operations should be performed in an expression. The script mentions the importance of understanding the order of operations when substituting inputs into a function's equation, especially when dealing with operations like squaring a number before multiplying it, as seen in '2x^2'.

💡Absolute Value

Absolute value in mathematics is a function that measures the distance of a number from zero on a number line, regardless of direction. It is denoted by vertical bars, as in '|3x - 5|'. The script explains how to handle absolute values in functions, such as 'G(x)', where the absolute value is calculated first before proceeding with other operations.

💡Polynomial

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication, and non-negative integer exponents of variables. In the video, 'f(x) = 3x^2 - 2x + 1' is an example of a polynomial, specifically a trinomial because it has three terms. The script uses polynomials to demonstrate how to apply function notation with equations.

💡Function Machine

The term 'function machine' is used metaphorically in the script to describe the process of a function taking an input and producing an output. It helps visualize the concept of a function as a 'machine' with an input 'chute' and an output result. For example, when the input 2 is put into the 'F machine', the function's inner workings calculate and output '-5/2'.

💡Simplifying Expressions

Simplifying expressions involves combining like terms and reducing an expression to its simplest form. The script demonstrates this process when dealing with the function 'f(x)' with the input 'x + 1', resulting in '3x^2 + 4x + 2'. The explanation shows how to square the binomial, distribute, and combine like terms to simplify the expression.

Highlights

Introduction to function notation with equations, emphasizing the importance of understanding the function as a machine with inputs and outputs.

Explanation of the generic formula to calculate outputs for any input in function notation.

Clarification on the role of the variable 'X' in function notation, representing any input value.

Demonstration of calculating the output of a function using a specific example, f(2).

Emphasis on the order of operations and the importance of parentheses in function notation.

Result of the example calculation, showing f(2) equals -5/2.

Introduction of additional functions f, g, and h with different algebraic expressions.

Explanation of absolute value notation and its application in the function g.

Example calculation of f(-5), illustrating the process of substituting the input into the function.

Result of f(-5) calculation, showing the output as 86.

Example calculation of g(-6), including the handling of absolute values.

Result of g(-6) calculation, with the output being -42.

Explanation of the constant function h, which always outputs -17 regardless of the input.

Example of applying function f with a variable input (X + 1), showing the process of substitution.

Result of the variable input example, where f(X + 1) results in the expression 3x^2 + 4x + 2.

Encouragement for viewers to practice applying functions with variable inputs and expressions.

Transcripts

okay so so far we've looked at function

notation but we've done it very visually

with mappings or sets of ordered pairs

or graphs now we're going to look at

maybe the most common application of

function notation and that's with an

equation okay so let's let's recall

first of all function notation you you

write your function name maybe it's F or

G or H whatever F of your input X and

that equals y so f of your input equals

your output okay

F of your input equals your output so

let's let's assume that we have some

function defined using this equation

okay well we'll do some analysis of this

so let's assume that f of X equals a

fraction two x squared minus three over

X minus four okay now this is a formula

to define this function it's not not a

visual representation to tell you that

well this input has this output it tells

you a generic formula to calculate

outputs for any input okay so so what

this is saying is that the way you use a

formula is the you change the value of

the variable to be whatever you want and

then that variable takes that value

everywhere so our variable here is our

input X okay and so the key is anyway

there's an egg's that's the variable

part of this equation that's the

variable f is the function name that

doesn't change right that's not a

variable that's the name of this machine

for which we can put inputs in and

receive outputs so whatever this X is

both of those X's are the same okay

those are both the same so so you can

kind of kind of think of this as think

of it as

don't get hung up on X don't get hung up

on X we use X because that typically

represents input in algebra but just

think of it as think of it as a machine

f with this chute right this is lit just

like a hole to the machine and you can

put whatever you want into that hole

right whatever input you want whatever

input you want put into this chute which

is going to feed it to the F machine and

it's going to output remember F of this

input whatever that happens to be that

whole thing is what your output is and

so the output of this is going to be two

times the input whatever it is squared

minus three over the input minus four

okay don't don't get hung up on X okay

it just means input and it can change

it's a variable so whatever whatever you

feed into the F chute it's going to come

over here into both of these places and

form a calculation to give you a final

final number output now let's do a

particular example with this function

where we actually calculate an output so

so let's do an example let's let's see

if we can determine the value of F of

two and so what this means is F of 2

means Y value means output so what we

need to do is since two is in

parentheses we know that's the input we

need to input 2 to this function by the

way whenever you see f of X equals like

this whenever you see that that is the

definition of the function so in this

case it's the definition of F it's very

important to understand that this

defines your function just like a

picture of a mapping defines what the

function does to an input same thing

here this is how you calculate outputs

for the function f so f of 2 means plug

2 in for X right and that means I have

to replace all of my exes with two

that's my input or 2 goes into the chute

as the input to the F machine

and then it goes in these spots in my

output now it's important to know your

order of operations when you do this so

for example when I say two x-squared you

have to understand what that means it

means you square X whatever it happens

to be whatever number you're plugging in

you square that first and then you

multiply it by two that's why the

parentheses are extremely important here

when you plug in an input if you're not

sure about the order of operations put

parentheses around your input every time

and that should help okay so if we do

that here it looks like two times two

squared minus three over two minus four

and then using the order of operations I

already mentioned if I square 2 I get 4

4 times 2 is 8 8 minus 3 is 5

so my numerator is 5 and I'm bottom 2

minus 4 is negative 2 so 5 divided by

negative 2 is negative it's a negative

fraction negative five-halves so that's

typically how we write that you could

write negative 2.5 often fractions are

preferred improper fractions not mixed

numbers now what does this mean this

means that if we take the input of 2 and

we input it right 2 is the input to the

F function or the F machine think of

this as a machine that has inner

workings if we input 2 to that and 2

goes through these inner workings all of

this happens and the machine finally

outputs negative five-halves ok so

that's kind of a visual way to think of

this and that's how you calculate

outputs for any input with this

particular function now let's practice

some more function notation with

equations and we'll try to throw in some

examples of commonly missed algebraic

calculations

and that's kind of what's tricky about

making a video like this is I can't

cover all the possible calculations you

you should know how to do so so I'm

gonna try to take on the ones that a lot

of people have trouble with or miss

frequently so here are some functions f

of X let's say is 3x squared minus 2x

plus 1 it's a polynomial it's a

trinomial actually because it has three

terms lots of different things we could

say about this G of X is negative 2

times a vertical bar and inside the

vertical bars I have let's say 3x minus

5 so I've got 3x minus 5 inside these

vertical bars we'll talk about those in

a minute plus 4 so do you remember what

the vertical bars mean in mathematics in

this particular case it means what's

called absolute value so we'll see how

to handle that in some cases if you

don't remember absolute value you might

want to go do some reading up on it

alright last let's say we have H of x

equals negative 17 such a simple

function but because it's so simple it

can be confusing

let's do examples now let's consider

let's consider f of negative 5 so that

means negative 5 is input to the F

function which when it's the equation

that means well here's X that means

these X's have to match that X so I need

to replace these X's with negative

negative 5 so we've got 3 times negative

5 quantity squared minus 2 times

negative 5 plus 1 well this is 3 times

25 that's 75 minus 2 times negative 5 is

negative 10 so I've got minus negative

10 which of course is plus positive 10

and then plus 1 so guess 75 plus 10 plus

1 which is 86 so if we input negative 5

to the F

function it outputs 86 if X is negative

5y is 86 for the F function okay so

that's one example with the F function

let's do another let's look at G of

negative 6 so now I go to my G equation

and I substitute negative 6 in for my

input so this X also turns into negative

6 it's about negative 2 times the

absolute value of 3 times negative 6

minus 5 close the absolute value plus 4

so we have absolute value we treat it

kind of like parentheses we figure out

everything inside first so 3 times

negative 6 is negative 18 negative 18

minus 5 is negative 23 keep your

absolute values we haven't dealt with

those yet and keep everything else as

well so now the next thing I need to do

is I need to multiply a negative 2 by

whatever this is so I need to know what

the absolute value of negative 23 is and

the absolute value of negative 23 is

positive 23 again you use your order of

operations so now this turns into

negative 2 times so my absolute value

bars turn into parenthesis negative 2

times positive 23 plus 4 so now we've

got negative 46 plus 4 is negative 42 so

G of negative 6 equals negative 42 if we

input negative 6 to the G machine the

output is negative 42 all right let's do

another one let's go to the H function

let's look at H of four point one that

means my input or my x-value is four

point one so any place there's an X I

replaced with four point one here the

output does not involve X at all it's

completely independent of your choice of

X so no matter what X is

the output here is negative 17 and it's

that simple

okay so H of 4.1 equals negative 17 and

no matter what you plug into this

function your output is negative 17

because X is not involved in the

calculation at all all right let's do

three more quick examples applying each

of these functions with a variable input

okay let's look at let's look at F of

now let's say the input is X plus 1

don't don't be scared by this all this

means is the input is X plus 1 meaning I

look at my definitions Y go to my F

definition which is right here and I

replace all of the X's with X plus 1 so

this turns into 3 times the quantity X

plus 1 squared minus 2 times the

quantity X plus 1 right I'm replacing

both of those with X plus 1 plus 1 and

then I just simplify this now I'm not

gonna get a number as my answer my

answer is going to involve X because my

input involved X as well all right so

the first thing I need to do is square X

plus 1 which that means X plus 1 times X

plus 1 if you need to write that out X

plus 1 squared means X plus 1 times X

plus 1 and you should be able to

multiply that pretty easily but when you

do you get x squared plus 2x plus 1 you

don't remember how to do that you may

need to do some research on X plus 1

times X plus 1 you just simply

distribute all of this out ok then here

I've got minus 2 times X plus 1 so I

need to distribute my negative 2

and then bring down my plus-one all

right and then we're in the end here we

have one more step well distribute our

three and I'm going to go ahead and

combine like terms so I've got three x

squared

that's my only x squared term then I've

got 6 X but then here I've got -2 X so

that's plus 4 X and then last I got 3

but then I've got minus 2 which is so 3

minus 2 is 1 1 plus 1 is 2 and I can't

simplify it any further I've gotten rid

of all my parentheses and combined all

the like terms and so if I input X plus

1 to this function f the output is 3 x

squared plus 4 X plus 2 okay we could do

the same sort of thing with the other

two functions in fact I think that's

good enough you should be able to input

numbers to these functions or

expressions right with variables

themselves and it's no big deal it's the

same process just plug in whatever this

is plug it in as your input which means

replace all the X's with that expression