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in this episode you’ll learn how to add and subtract functions we’ll also

practice multiplying and dividing them let’s do some math [Music]

hi everyone welcome back so we’re going to begin by talking about the sum

difference product and quotient functions so here’s the sum

and here’s the difference and so for example how these work is

we’re going to be given two functions f and g

and so we know what the domains are a and b so f has domain a

and the function g has domain b and we’re going to be defining a new function

based upon these two given functions here so

f plus g is a new function now in order to define this function right here

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so we’re going to need to specify the inputs and outputs so the way that we do

this is that we’re going to give f plus g and input

and so let’s give it an input so let’s say our input is x

and then we’re going to specify how to find that output

so the way to find the output from f plus g function is we input x into f

and we input x into g and then we add those two real numbers

together and then we get the output so that’s the sum function so we put x into

the sum function by substituting it into f into g and then adding those two real

numbers together so in order for this function to work f plus g

x needs to be in the domain of f and x also needs to be in the domain of

g if you try to input a real number into the sum

and that real number x isn’t in the domain of f well then you have no way to

get an output so this x has to be in the domain of f and the domain of g

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and the same thing for minus the only difference is we’re getting the two

outputs and then we’re subtracting this in this order here f f of x minus g of x

for f minus g and so then we’ll look at f times g

which is f of x times g of x so there’s implied multiplication right here

and then we’re looking at the last function right here f the quotient of f and g

to input an x into this function we simply take the outputs here and divide

now that of course assuming that not only is x in the domain of f and x is in

the domain of g but also g of x here is non-zero so we can say

that right here so the input is in the intersection has to the input has to be

in a and has to also be in the domain b and we say that this denominator here is

non-zero so then we have the quotient defined so this right here means i’m

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defining this new function right here using what’s already already known over

here on the right so let’s look at lots of examples so

you know just for example let’s say here we have f of x is x squared minus 5x

and let’s say g of x here is 2x minus 9 and so what will these four functions

right here be so let’s find the summation function

and let’s do that right here so f plus g evaluated at some input x

is going to be uh f and then plus the g of x so plus g of x and this is

all g of x right here 2x minus 9. all right so very good so this what does

this come out to be x squared and then we have a minus 3x here

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and then we have a minus 9. and so we can substitute in now we can

substitute in values right here for example f plus g at say 1

will be what 1 squared minus 3 minus 9. so what’s that minus 11 right and so we

can find these other four functions very similarly so what is f minus g of x

this will be f of x all of that minus the g of x and so this becomes x squared

and then we have minus five x minus two x so that’s minus seven x and we have

minus minus so we have a plus nine so this is the difference between f and g

this is the difference function notice that that is different than g of

g minus f of x so we could go find this function as well

but that but these two functions are not the same so this would be g of x

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minus f of x and so this would be 2x minus 9 minus and then f of x

and so f of x is x squared minus 5x and so this would be minus x squared

and then this would be a positive 5x along with the 2x so that’s 7x

and then we have minus 9 here so you can see the relationship between

these two functions right here they’re not the same and they are different

um so what about uh the product function we can find the product function right

here f times g is f of x times g of x which is just x squared minus 5x times the

2x minus 9. and if you wanted to we could expand that out and simplify that

all right and so what would the quotient function be here

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now before i say that what is the domain and range of these so far that we’ve

done so far so the domain of f here is all real

numbers and the domain of g is all real numbers

and so the intersection is just going to be all real numbers in other words

i can input any x i want into the summation any real number here because i

can input any real number into f and i can input any real number to g

and well we can always add any two real numbers together to get another real

number so there’s not going to be any restriction upon the domain for f plus g

and similarly for f minus g and f times g

there’s no restriction the domain is all

real numbers for these two so the domain is going to be all real numbers for

these three also we can substitute in any x we want into

here and we’ll get an output so let’s look at f divided by g now the quotient

so let’s just make a little room right here and

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so here’s f the quotient of f and g evaluated at some input and so by

definition this is just f f over g and so we get f of x

and then over 2x minus nine and so this is the

quotient function right here this is this this is a rational function

notice here that because we’re dividing uh by 2x minus 9

that the domain here is not all real numbers so the domain is all real numbers

such that x is not equal to whatever makes this denominator 0 if

anything so it turns out that 9 over 2 makes the denominator 0. so 9 over 2.

so this is the domain all real numbers except 9 over 2. okay so there’s uh

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an example of using these four right here and let’s perhaps look at some more so

let’s say we have evaluate the sum difference product and quotient

so let me give you some more let’s get some more functions going up here

so now let’s say we have a function here f of x is um x squared plus one

and our g of x is x minus four all right and so now let’s look at some

different things that we can do so let’s find so for part a here we’ll

say f minus g of zero let’s try to find that and f minus g at three t

and f times g at six and f over g at five and for the last part let’s say f

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over g evaluated at minus one minus g of three

so let’s find uh all of these right here giving these two functions right here

all right so let’s do that all right so let’s find f minus g at

zero now there’s two ways that we can do this we can find

this function right here for all input for all input x in other words we can

find a formula for it and then we can go plug in x into that formula right there

but you know we don’t really need to do that it’s just f of 0 minus g of 0

just by definition of what the difference function is

it’s f of 0 minus g of 0. so now i can just go plug in 0 into each one of these

so if i plug in 0 here i get 1 minus and then i plug in g i plug into 0 into g

and so i get minus four and so this is just one plus four or in other words five

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all right what about f minus g of three t so this is f of three t

minus g of three 3t and let’s see here what is f of 3t so

i’m going to put in 3t here so this would be 9t squared plus 1

and then a minus sign right here in fact i’ll put parentheses just to

help with the form and then now g of 3t so g of 3t so i need to put 3t in here

so it’s 3t minus 4 and we can carry this out a little bit so nine t squared

and then minus three t and then let’s see here we have plus five

all right looks good yeah plus five all right so now let’s do f

times g at six so this will be f of six times g of six and f of six is what um

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36 plus once that’s 37 and then g of two is sorry g of 6 is 2

so that is 37 times 2 which is 4 carry the 1

74 right and so let’s look at f of 5 divided by g of 5.

now if g of 5 is 0 then we’ll have to erase this and say does not exist

but let’s see if we can do it so f of five is 25 plus one so 26 over

and now let’s see if we get zero here or not

g of five so five five minus four is one so we’re good shape if this turned out

to be zero for example if they had asked us what is what is this at four in fact

i’ll just do that over here let’s put a 5 back in here that’s just

- what if they asked us to find g of 4 sorry f over g at 4

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so this would have been 16 plus 1 which is 17 and then this would be zero

so you know you wouldn’t leave your answer like this this is not something

you would write down because this this this is not a number

so you’re not going to be able to use it with an equal sign that’s not the way

the equal sign makes any sense so you cannot write this down so what we’ll

just say is this right here is is undefined it’s not defined

all right so now let’s do this one right here so f over g at minus one

and so that’ll be f of minus one over g at minus one and then minus g of three

all right so i just want to write it out a little bit better here now let’s go

substitute everything in so g at my f at minus one so that would be a two

and then g at minus one will be minus five

minus and then g at three which will be another minus one so that’ll be you know

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minus two over five plus one which will be five over five

so that’ll be three fifths all right looks good

all right so let’s see what else we can do let’s use graphs to evaluate now

yeah let’s use graphs to evaluate so i’m going to give you some graphs and

we’re going to have to evaluate now instead of giving rules so let’s

let’s do something like that so let’s write a graph for f

and so it’s going to be shaped like a v it’s going to come down and then go back

up and this is going to be two this is going to be about a one here

about a three and then about a four um and the height here is a four

three and then two um and then a one let me see if i can space them better

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so let’s say one two three all right that’s good enough and

so now let’s say here we have a uh g over here and let’s just say this one goes

straight down let’s put this out of 4 also

and i’ll say there’s a three and a two and a one right there

and then this is a four and then halfway is a two and then a

three and then a one all right there we go that’s good enough

there’s an f and there’s a g and so let’s evaluate some

combinate function combination so let’s do f plus g at three

and let’s do f minus g at one and let’s do f over g at two and then

we have room for one more here let’s do f times g at four so let’s find these

let’s evaluate these functions right here in these places right here so f plus g

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at three right so that’s f of three plus g of three what is f of three

f of three here is three g at three is one so that’s a 4.

and so f of 1 minus g at 1 so f of 1 is 3 and g at 1 is a three so this is zero

f over g the quotient here that’s f of two divided by g of two if possible

so f of two is 0 so that’s 0 over g of 2 is not 0. it’s a 2 right here

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so 0 over 2 is 0. and then here we have f of 4 times g of 4 f 4 is 4 g of 4 is 0

so this one is also 0 here so sometimes you might have a graph you

can evaluate these functions right here without having to go actually make a

whole graph of these right here if you’re just looking to evaluate these

functions right here we can just evaluate them right here

but you know what let’s go ahead and use uh some functions now and let’s change

this up here let’s go using graphs to evaluate now let’s use using graphs to

graph let’s and let’s pick on the sum right here so let’s do something like

that real quick let me get this out of the way real quick

and here we go so i’m going to give you um this um

yeah so here we go so i’m going to give you an f and a g and then we’re going to

graph the sum let’s see if we can do this so here’s going to be a function right

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here i’m going to need a 1 2 3 1 2 3 and i’m going to need a 2 about here

and a 4 about here so i’m going to go and let’s say one’s about here

so i’m going to go here up to right there

and then i’m going to come down to about right there

and this is going to be the graph of my f so this is going to go up to about 2

right there so this is a graph of f now let’s look at the graph of g

so g is going to go from right here that’s going to be a minus 1

to 0 right there and it’s going to go [Music]

up right here also and hit right there and then it’s going to come back down

and hit right there okay so there’s g right there it’s just

coming straight up right there it goes through one

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and then it comes back down and goes right there so that’s that’s g now to

see g are different from f let’s put g in red right here so there’s g right

there and it’s coming down through there so there’s g i’ll put g right there

all right so maybe that’s hard to read maybe i can just

make that a little bit redder and this part here a little bit redder

and so yeah there we go so let’s look at what would the graph of f plus g look

like all right so let’s do that over here so um let’s find f plus g at

zero f plus g at one f plus g at two and f plus g

at three let’s find these right here and then connect them with lines so what

we’re going to do is say f plus g so here’s f and g right here so what is f

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of 0 right that’s f of 0 plus g of 0. so f of 0 is 2 and g at 0 is minus 1

so that comes out to be a one and what is f plus g at one it’s three plus zero

right so f of one is three and then g of one is zero so that gives us three

and then f of two plus g of two so f of two is one and g of two is one

so it’s one plus one is two and this one right here f of f of 3 right here is

is so f is coming right through here f bounces at 2 and

comes back up so that’s f right there so f of 3 is 2 and g at three is zero

so this is also two so at zero we’re going to be at one so

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at zero we’re going to be height of one and at one we’re gonna be a height of 3

and then at 2 we’re going to be a height of 2.

well i kind of made that look messy all right and then at 3 we’re still at 2

so it’s constant right there that kind of makes sense because

when we pick a number between two and three say like this one right here

we’re a little bit short of two but how much short of two are we that part right

there so if you combine those together and add them together so this is the

graph of f plus g all right so that looks good all right so

let me know if you guys had fun in this video let me know if you want to see

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some more examples i look forward to seeing you

in the next video and i’ll see you then have a great day

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