Question

The Martians ask you to explain one last thing, Ultimate Math Ambassador. They ask you to create a new function, h(x). Then assign any number to x. Using complete sentences, explain whether f(h(x)) and h(f(x)) will always result in the same number. You will use the function f(x) that you created in problem number 2.

Answer 1

SOOOO LOST

Answer 2

what was the f(x) you got in 2?

Answer 3

@myininaya I'll post what I have so far as a file.

Answer 4

so you said f(x)=10x+3
so have you create h(x) yet?

Answer 5

oh so you have h(x)=2x+10

Answer 6

Yes and I'm stuck, I'll post the actual question also

Answer 7

you found h(4)
now find f(18) using f(x)=10x+6

Answer 8

The year is 2033, and intelligent life has been found on Mars! Little green Martians have found their way to Earth and have appointed you as the Ultimate Math Ambassador to their planet. You must help them understand how Earthlings use math.

Two Martians, Splott and Fizzle, have solved the equation 2x + 4 = –3x + 14. Examine the work of Splott and Fizzle. Identify any errors in the Martians’ calculations and explain, using complete sentences, what corrections they should make.

Splott Fizzle
2x + 4 = –3x + 14 2x + 4 = –3x + 14
2x + 4 + 3x = –3x + 14 + 3x 2x + 4 – 2x = –3x + 14 – 2x
5x + 4 = 14 4 = 14 – 5x
5x + 4 + 4 = 14 + 4 4 – 14 = 14 – 5x – 14
5x = 18 –12 = –5x
5 x divided by 5 is equal to 18 divided by 5 –12 + 5 = –5x + 5
x = 18 over 5 –7 = x
Create your own function to teach the Martians about functions. Your function must contain at least two different operations.
Using complete sentences, prove to Splott and Fizzle that your function is a legitimate function.
Using your function, explain to the Martians how to solve for f(3). Show your work and explain each step using complete sentences.
Using complete sentences, describe to the Martians how to find the inverse of your function.
The Martians ask you to explain one last thing, Ultimate Math Ambassador. They ask you to create a new function, h(x). Then assign any number to x. Using complete sentences, explain whether f(h(x)) and h(f(x)) will always result in the same number. You will use the function f(x) that you created in problem number 2.

Answer 9

I'm just stuck on the last part

Answer 10

pick a random value for x
and you did which was 4
you evaluated h(4)=2(4)+10=18
but you need to evaluate f(h(4)) and h(f(4))
basically it looks like they want you to test the commutative property over operation composition

You found h(4)=18
so what is f(h(4))=f(18)?

Answer 11

So, if we substitute f(x) for f(18) it would work out like this:
f(18)=10(18)+6= 180+6=186
???

Answer 12

right

Answer 13

now you find h(f(4))

Answer 14

and you want to see if f(h(x))=h(f(x)) for any value x.

Answer 15

So which equation would I use? h(x)=2x+10 and substitute 18 for x?

Answer 16

@myininaya

Answer 17

you are still using the same x value which is 4

Answer 18

you are trying to see if f(h(4)) is the same as h(f(4))

Answer 19

so you found f(h(4)) which was 186
now let's find h(f(4))
first find f(4)

Answer 20

f(x)=10x+6
f(4)=10(4)+6
=46
I'm sorry, I'm horrible at math

Answer 21

ok now we find h(46)

Answer 22

h(x)=2x+10
=2(46)+10
=102

Answer 23

@myininaya

Answer 24

right so clearly f(h(x)) does not equal h(f(x)) for all x.
What would we need in order for f(g(x)) to equal g(f(x)) for all x over the domain of f composed of g or g composed of f?
what do f and g have to be?

Answer 25

I literally do not have a clue, how did g come in? Lol
I feel so dumb

Answer 26

I was just introducing a new function
and it was also a hint

Answer 27

because you used g somewhere on your paper

Answer 28

Right, as the inverse

Answer 29

so if f and g are inverses then f(g(x)) will always equal g(f(x)) for any value x right?
but h was not the inverse of f so f(h(x)) will not always equal h(f(x)) right?

Answer 30

right

Answer 31

@myininaya

Answer 32

yes?

Answer 33

right

Answer 34

So I'm done then, right? Because the last question was if they would always equal and the answer to that is no.

Answer 35

right
they would equal for all x in the domain if f and h were inverses

Answer 36

but they aren't

Answer 37

or if the two functions were the same function :)

Answer 38

like f(f(x))=f(f(x)) for all x right?

Answer 39

yes

Answer 40

So, is all of this correct?

Answer 41

yeah looks good...
I would just change those 3 x's down there to 46 to show you are finding the value of h(46) and i know you are because you said you are but it just looks better
to say h(46)=2(46)+10
h(46)=92+10
h(46)=102
well and also close to the start of number 6 I would say h(4)=2(4)+10=8+10=18
those are the only changes to number 6 I would make.

Answer 42

and I guess you conclude somewhere after finding h(46)=102
that h(f(4))=102

Answer 43

Thank you soooo much! I would hug you if I could! I hate math lol

Answer 44

You did good.

Answer 45

You're awesome!