Question

composition of functions: f(x)=3x-2, g(x)=x²+1, find (f°g)(-1) and (f°g)(3). can anyone help me understand how to go about it?

Answer 1

fog means in f(x) put in g(x) in place of x

Answer 2

yes first of all get rid of the circles.
\[f\circ g(x)=f(g(x))\]

Answer 3

so
f(x) = 3x-2

and so
f(g(x)) would be 3g(x) - 2

Answer 4

now put in the definition of g(x)

Answer 5

I'm so lost.

Answer 6

then work from the inside out. start by replacing (x) by what it actually is in this case
\[f(g(x)) =f(x^2+1)\] and now replace the x in f(x) by
\[x^2+1\] to get
\[3(x^2+1)-2\]
\[3x^2+3-2\]
\[3x^2+1\]

Answer 7

ok lets go slow

Answer 8

lets start with
\[f\circ g(-1)\]

Answer 9

what is
\[g(-1)\]?

Answer 10

is it clear how to find
\[g(-1)\]?

Answer 11

no its like I'm looking at a foreign language. I don't have any idea what I am supposed to be doing

Answer 12

ok so lets do that first

Answer 13

\[g(x)=x^2+1\] so if i want to find g of some number, i replace x by that number

Answer 14

Yes functions I understand.

Answer 15

\[g(-1)=(-1)^2+1=1+1=2\]
\[g(2)=2^2+1=4+1=5\]

Answer 16

ok so let me ask again, what is
\[g(-1)\]?

Answer 17

oh actually i wrote it for you.


\[g(-1)=2\] yes?

Answer 18

im a lost cause i guess

Answer 19

so the only additional step to finding
\[f\circ g(-1)\] is finding
\[f(2)\] because that is what it means. first find g(-1) then find f of the result

Answer 20

lets go back for a second.
\[g(x)=x^2+1\] what is
\[g(3)\]?

Answer 21

10

Answer 22

of course this is not a lost cause. once you get it you will be like 'oh is that all?'

Answer 23

right. now what is
\[f(10)\]?

Answer 24

i forgot what f was
\[f(x)=3x-2\]

Answer 25

28

Answer 26

so now the question is what is
\[f(10)\]

Answer 27

got it!

Answer 28

f(g(x) = f(x^2 +1) = 3(x^2+1) -2 = 3x^2 +1
fog(-1) = 3 (-1)^2 + 1 = 4
fog(3) = 3 *3^2 + 1 = 28

Answer 29

so now we just have to understand what
\[f \circ g(3)\] means. first you found that
\[g(3)=10\] then you found that
\[g(10)=28\]
you are done.

Answer 30

sorry i meant
\[f(10)=28\]

Answer 31

typo there. so the idea is
a) first find g(3)
b) then find f of the result. you have it

Answer 32

but that's not the answer I should have. my text book tells me that it should be g(-1)=4 and f(3)=50

Answer 33

hold on

Answer 34

\[g(x)=x^2+1\] yes?

Answer 35

no, first you calculate the fog formula and THEN the fog(3). etc

Answer 36

yes

Answer 37

ok so there is some problem with your text book not with you. unless you are looking at the wrong question

Answer 38

it shows me step by step how it is solved. I'm not wrapping my mind around it though thats why I'm here

Answer 39

because if
\[g(x)=x^2+1\] then i guarantee you that
\[g(-1)=(-1)^2+1=1+1=2\] you can bank on it

Answer 40

first off we have to agree that
\[g(-1)=2\] yes?

Answer 41

yes

Answer 42

no matter what the text says.

Answer 43

then it puts the 2 into the fog problem: 3(2)-2=4

Answer 44

so now we want to compute
\[f \circ g(-1)=f(g(-1))\]

Answer 45

f(3) is where the problem is different

Answer 46

not sure what you mean.
\[f(3)=3\times 3 - 2=9-2=7\] but you are not asked for
\[f(3)\]

Answer 47

looks like you are asked for
\[f(g(3))\] unless that is a typo and you want
\[g(f(3))\]

Answer 48

if the problem is written f(x)=3x-2, g(x)=x°+1 find fog (-1) and fog (3). why does it seem to be inputting them backwards? why does g(x) get he -1 and not the 3

Answer 49

aaaaaaaaaaaaaaaahhhhhhhhhhhhhhhhhhhh

Answer 50

no g(f(3)) doesn't give 50 i tried this

Answer 51

they are giving you the answer to
\[g(f(3))\]

Answer 52

no it is not 50

Answer 53

i feel so stupid why cant i understand this

Answer 54

\[f(3)=7\]
\[g(7)=7^2+1=50\]
so
\[g(f(3))+50\]

Answer 55

you are not being stupid the text is wrong!

Answer 56

either the text is wrong or the question are written wrong.

Answer 57

lets make sure we have the problem correct with f and g correct as well.

\[f(x)=3x-2\]
\[g(x)=x^2+1\] for sure yes?

Answer 58

yes that is correct

Answer 59

and not the other way around

Answer 60

ok. and for sure you are asked for
\[f\circ g(-1)\] not
\[g \circ f(-1)\] yes? not the other way

Answer 61

fog(-1) gof(3)

Answer 62

ok
\[f\circ g(-1)=f(g(-1)=f(2)=3\times 2-2=6-2=4\]

Answer 63

ok now i see the problem. second question is
\[g\circ f(3)\] not
\[f\circ g(3)\]

Answer 64

right

Answer 65

they are different. for
\[g\circ f(3)\] you need to compute
\[f(3)\] first

Answer 66

so first you compute
\[f(3)\] and get
\[g(3)=3\times 3-2=9-2=7\] yes?

Answer 67

yes

Answer 68

damn that was a typo. i mean
\[f(3)=7\]

Answer 69

ok and now you compute
\[g(7)\] to get
\[g(7)=7^2+1=49+1=50\] yes?

Answer 70

yes

Answer 71

whew. so we have
\[g\circ f(3)=g(f(3))=g(7)=50\] and now the answers are correct and the same as in the text yes?

Answer 72

yes

Answer 73

idea is this
\[g\circ f(x)\] means first do f, then do g whereas
\[f\circ g(x)\] means first do g, then do f.

they are different and almost always give different answers

Answer 74

OMG that is the first thing to totally make sense to me

Answer 75

its backwards (to me) but I can work with that

Answer 76

that is because we read from left to right but we read functions the other way, from right to left

Answer 77

you would thing
\[f\circ g\] means first f then g because that is the way we read, but it does not

Answer 78

the x is on the right
we read
\[f\circ g(x)\] means first
\[g(x)\] then
\[f(g(x))\] in other words first g, then f

Answer 79

ahh. my text doesn't explain anything at all. I'm taking my class online so I cant raise my hand in class either

Answer 80

thank you so much.I think I can try a few on my own

Answer 81

would you mind checking my work on one?

Answer 82

ok you can give me an example if you like
i will be back in 5 minutes. i'll leave this open

Answer 83

ok thanks

Answer 84

just post it here

Answer 85

f(x)=x°-2, g(x)=x+4 find fog (2) and gof (-4)
g(x)=x+4
g(s)= 2+4=6

f(-4)=(-4)°-2
=16-2=14

fog(6)=f(g(6))
=6°-2=36-2=34

gof (14)=g(f(14))
14+4=18

Answer 86

ok let me check

\[f(x)=x^2-2\]
\[g(x)=x+4\]
\[f\circ g(2)=f(g(2))=f(6)=6^2-2=36-2=34\]

Answer 87

looks good to me!

Answer 88

\[g\circ f(-4)=g(f(-4))=g(14)=14+4=18\]

Answer 89

that one looks good too!

Answer 90

wow i think I understand it

Answer 91

you are a lifesaver

Answer 92

whew. not to say "i told you so" but
\[\color{red}{\text{i told you so!}}\]

Answer 93

not that hard once you know what you are doing. now before you go

Answer 94

lets make sure you can also compute
\[f\circ g(x)\] in the above example

Answer 95

lol. thank you thank you thank you!!!!

Answer 96

we do it the same way, but with "x" instead of numbers.

Answer 97

\[f\circ g(x)=f(g(x))=f(x+4)=(x+4)^2-2\]

Answer 98

and you see if you replace x by 2 in the above you will get
\[f \circ g(2))=f(g(2))=(2+4)^2-2=36-2=34\] just like before