Question

Write the given function as the composite of two functions, neither of which is the identity function, f(x) = x
http://media.education2020.com/evresources/649-03-02-00-00_files/i0270001.jpg

Answer 1

A composite function is just one function contains another function

let s(x) and g(x) be separate functions.

Notation of composite functions is,

(s o g)(x) is the same as s(g(x))
(g o s)(x) is the same as g(s(x))

for example

given s(x) = 1 + x
and g(x) = x^2

(s o g)(x) = s(g(x)) = 1 + x^2
and
(g o s)(x) = g(s(x)) = (1 + x)^2

Answer 2

you should be able to solve this now

Answer 3

This is actually good practice if you ever get into calculus, as it can make taking derivatives a lot easier.

Answer 4

I'm actually really confused lol. My teacher just threw this at me. But thanks, imma sit here and give some thoughts on it hahaa

Answer 5

It is pretty simple actually dont fret,

look at my example,

(g o s)(x) = g(s(x)) = (1 + x)^2

see a similarity between this composition and your function?

Answer 6

hint:

\[\sqrt{x} = x^{\frac{1}{2}}\]

Answer 7

\[\sqrt[3]{x} = x^{\frac{1}{3}}\]
essentially,

\[\sqrt[n]{x} = x^{\frac{1}{n}}\]
where n is just a number

Answer 8

just to explain this notation just incase you didnt understand it

Answer 9

I will do a few more examples maybe that will help,

let,

g(x) = 2 and f(x) = x

then,

(f o g)(x) = f(g(x)) = 2

another example

let c(x) = x^(1/2) and d(x) = 1 + x

then,

(c o d)(x) = c(d(x)) = (1+x)^(1/2)

Answer 10

its like subing a number in but instead you are subing in another function

Answer 11

You just need to think about how this function can be split into two functions

Answer 12

My last example is very very similar to your problem

Answer 13

if you make a guess I can tell you if you are on the right track

Answer 14

Just a final note in this problem you are just doing the reverse you are starting wtih a function then devise what two functions one subbed into the other make it up

so,

f(x) = (g o s)(x) = g(s(x)) = (x^2 + 2)^(1/3)

we need to come up with what g(x) and s(x) are to answer this question

Answer 15

Okay so I don't know if I'm even doing this right
but I got something close to
f(x) = sqrt(x+2)
g(x)= x

but something is missing

Answer 16

omg no i forgot the about thr ^(1/3)

Answer 17

yes you did :)

Answer 18

You are close but I feel like a teacher wouldnt give you full marks because you can sub any function into g(x) = x or vice versa and get the same function back. you arent creating two distinct functions from the single function

(f o g)(x) = (x + 2)^(1/3)

Answer 19

you are very close though

Answer 20

think of it as an inner and outer function

Answer 21

look at my examples again briefly you are on the right track though

Answer 22

essentially change g(x) = x to the outer function and f(x) to the inner function

Answer 23

for example,

in the function

(x + 1)^5

x + 1 is the inner function
x^5 is the outer function

Answer 24

if you sub x + 1 into x^5 you get the function (x + 1)^5

Answer 25

i'm just confused on how to do the ^(1/3)

Answer 26

its just another exponent nothing special

Answer 27

wait so
(sqrt(x+2))^2 ???

Answer 28

if you did that,

\[\sqrt{x+2}^{2} = ((x+1)^{\frac{1}{2}})^2 = (x+1)^\frac{2}{2} = x + 1\]

Answer 29

f(x) = (x+2)^(1/3)
g(x) = x^2

Answer 30

you need to include the composite how are you subing one function into the other to get the one you are looking for

Answer 31

like (fog)(x) ??

Answer 32

if you did that would you get the function you are looking for??

Answer 33

Another example,
again, split into an inner and outer function

(5 + x)^(1/5)

we see that 5+x is an inner function
we see that x^(1/5) is the outer function

we sub the inner function into the outer function and we get (5 + x)^(1/5)

Answer 34

this example is pretty much exactly the same as your problem

Answer 35

whaaaaaaaaaaaaa :o

so it could also be
f(x) = ^(1/3)
g(x) = x^2 +2

????????????

Answer 36

yes!!!

Answer 37

g(x) is your inner function
f(x) is your outer function

Answer 38

whaaaaaaaaaaaaaaa :o
how am i doing this omg

Answer 39

thanks so much!!

Answer 40

I think you understand?

Answer 41

Yeah, I think I understand lol....

Answer 42

now write the composite

Answer 43

If the outer function is subbed into the inner function you will get the answer wrong, so you have to use composite notation

Answer 44

to show your final answer

Answer 45

(f o g) is right??

Answer 46

yes f(g(x)) is correct, I like to think of it as f is eating g to remember the notation where the circle is a mouth but I'm weird

Answer 47

yasssss ^.^
thank you!!