Question

You make a purchase at a local hardware store, but what you've bought is too big to take home in your car. For a small fee, you arrange to have the hardware store deliver your purchase for you. You pay for your purchase, plus the sales taxes, plus the fee. The taxes are 7.5% and the fee is $20.

Answer 1

(i) Write a function t(x) for the total, after taxes, on the purchase amount x. Write another function f(x) for the total, including the delivery fee, on the purchase amount x.
(ii) Calculate and interpret ( f o t)(x) and (t o f )(x). Which results in a lower cost to you?
(iii) Suppose taxes, by law, are not to be charged on delivery fees. Which composite function must then be used?

Answer 2

(i) Write a function t(x) for the total, after taxes, on the purchase amount x. Write another function f(x) for the total, including the delivery fee, on the purchase amount x.
(ii) Calculate and interpret ( f o t)(x) and (t o f )(x). Which results in a lower cost to you?
(iii) Suppose taxes, by law, are not to be charged on delivery fees. Which composite function must then be used?

Answer 3

did u understand wat they asking ?

Answer 4

Not really.

Answer 5

basically they want you write two functions : t(x) and f(x)
t(x) : represents cost x + tax of 7.5 %
f(x) : represents cost x + delovery fee of $20 flat

Answer 6

once you have the functions, working rest is bit easy..

Answer 7

Oh ok. Yeah..ok..
How would you write the functions tho?

Answer 8

say, the cost is x

Answer 9

ok

Answer 10

t(x) : represents cost x + tax of 7.5 %


t(x) = x + 7.5 % of x

simplify

Answer 11

is there a dif way to write tax of 7.5%?

Answer 12

7.5% of goods purchased = 7.5/100*x = 0.075x

Answer 13

overall,

t(x) = x + 7.5 % of x
= x + 0.075x
= x(1+.075)
= 1.075x

Answer 14

t(x) = 1.075x --------------------------(1)

Answer 15

similarly work the f(x)

Answer 16

Where did the 1 come from!?

Answer 17

t(x) = x + 7.5 % of x
= x + 0.075x
= x(1+.075) *******************
= 1.075x

Answer 18

you're asking that 1 ?

Answer 19

Yesh.

Answer 20

look at the line right above that :-

x + 0.075x

Answer 21

ive just factored out x

Answer 22

I don't get it..

Answer 23

Oh wait..

Answer 24

I GET IT!

Answer 25

Oh wait.. no I don't. Shouldn't the 2 x's stay?

Answer 26

:)

here it is

x + 0.075x
^ ^

Answer 27

we see that x exists in both terms, so we pull that out

Answer 28

x(1 + 0.075)

Answer 29

when we pull out x from first term, 1 stays there

Answer 30

Oh ok.

Answer 31

when we pull out x from second term, 0.075 stays there

Answer 32

So then would it become t(x)=1.0755

Answer 33

*t(x)=1.075

Answer 34

who ate x ? ;)

Answer 35

But..u pulled it out..and shemurr?!

Answer 36

x(1 + 0.075)
x(1.075)

1.075x

Answer 37

Oh wait.. i see..

Answer 38

we pulled it out , but it dint go anywhere. it stuck there

Answer 39

okay, once we make peace wid t(x) function,
t(x) = 1.075x ---------------------------(1)

Answer 40

save it, and put the rquation for f(x)...
once we have both funcitons, we can work the rest

Answer 41

*equation

Answer 42

f(x) : represents cost x + delovery fee of $20 flat

Answer 43

So would that be, x+20?

Answer 44

perfect !

f(x) = x+20 ---------------------------(2)

Answer 45

whats that 2 for?

Answer 46

im just labelling it as number 2, so that we can refer to that later
its just for communication between us

Answer 47

ah ok

Answer 48

So this is about as far as I got on the second step..

Answer 49

we're done wud part1,
ever dealt wid composition functions before ? fog etc...

Answer 50

Yeah I have. just word problems are not my thing lol

Answer 51

yeah who doesnt lol

Answer 52

f o t(x)=1.075x + 20
t o f(x)=22.575x

Answer 53

lol

Answer 54

fot(x) looks correct
check tof(x) again

Answer 55

t o f(x)
t(x+20)
t(x)=1.075x
t(x)=1.075x(x + 20)
= 1.075x^2 + 21.5x

Answer 56

that?^

Answer 57

Yup looks good :)

Answer 58

lets put all equations we go in one place

Answer 59

t(x) = 1.075x --------------------------(1)
f(x) = x+20 ---------------------------(2)
(fot)(x) = 1.075x+20 ---------------------(3)
(tof)(x) = 1.075x + 21.5 ----------------(4)

Answer 60

ur calculatioon, above has a mistake. see correct working below :-

t o f(x)
t(x+20)
t(x)=1.075x
t(x+20)=1.075(x + 20)
= 1.075x + 21.5

Answer 61

ohh ok. i gotcha now

Answer 62

So now it asks which equation out of the 3rd or 4th is lower

Answer 63

in cost

Answer 64

which that means the 3rd is lower

Answer 65

Sure ?

Answer 66

yes I am sure

Answer 67

put aside equations for some time

u purchased $100
Is having 7.5% tax first, followed by fee of $20 a better option,

or

Is paying $20 first, followed by 7.5% tax a better option ?

Answer 68

Ahh you're right actually i was thinking something else lol

Answer 69

Uuhm the tax first right?

Answer 70

lol ok

Answer 71

yyayyyyy!!!

Answer 72

haa u got it !!!! always, have tax on low amount... so that we wud have to pay less tax to govt

Answer 73

yeah haha

Answer 74

Ok now 3rd part

Answer 75

you can answer that easily.
it says, ur govt is bit generous and it wont charge tax on delivery fee,

so we must use which composition funciton ? 3 or 4 ?

Answer 76

the same right?

Answer 77

3rd

Answer 78

thikn of it like this :-
which function shows, tax is not done on delovery fee ?

Answer 79

yup !

Answer 80

we wud ude 3rd function, when govt is good, and we dont have to pay tax on delivery fee

Answer 81

ohh ok. THanks!

Answer 82

*use

Answer 83

Lol govt good.

Answer 84

np :) no govt is good i suppose... lol

Answer 85

we're still paying taxes

Answer 86

lolol

Answer 87

That world problem would be like a dream come true haha

Answer 88

lol yes tax free economy is possible oly in books

Answer 89

lol