Question

Inverse functions:
I'm having trouble figuring out what this means (in terms of which way to begin solving):

f(x) = 10x + Sqrt(x+101)

what is (f^(-1))'(0) =?

that's (d/dx)(inverse)(0).. I don't get it.. do you find the inverse of f, then the derivative and then plug in zero? If so can you please show me how you solve this? Thank you!

Answer 1

ok lets make sure you are not confused in to thinking you can actually find the inverse of this
you cannot, so don't try it

Answer 2

what you are going to use is that
\[\frac{d}{dx}[f^{-1}(0)]=\frac{1}{f'(f^{-1}(0))}\]

Answer 3

ok, i was trying.. thanks for telling me this

Answer 4

as you see from the above, you do not need the derivative of the inverse, you need only know two things:
the inverse of 0, and the derivative of the original function \(f\) evaluated at that number

Answer 5

so first lets find the inverse of 0 by inspection
\[f^{-1}(0)\] is the number you would plug in to \(10x+\sqrt{x+101}\) to get \(0\)
what do you think that number is? (hint, don't do algebra, just think)

Answer 6

still thinking?

Answer 7

yes, because the only way i can think of doing this is with algebra.. and im forcing my self to not htink that way.. this is bad haha

Answer 8

im goign to just say -101..

Answer 9

or 0

Answer 10

oh heck no
0 doesn't work because if \(x=0\) you get \(10\times 0+\sqrt{101}=\sqrt{101}\) think again

Answer 11

10x^2 +x+101.. and just solve for x?

Answer 12

lol.. ok too much pressure.. hold on

Answer 13

nope

Answer 14

think that is all
what would make this beast zero?

Answer 15

oh crap.. it looks much more complicated.. like (10x + sqrt(x+101))^2

Answer 16

it is not supposed to be hard
you have \(\sqrt{x+101}\)
what would you put for \(x\) to make a very simple square inside there?

Answer 17

stop doing algebra

Answer 18

i can tell you are doing algebra
way too much work
suppose i asked you what would make \(2x+\sqrt{x+5}=0\) would that be easier?

Answer 19

kind of.. i feel like you would take 1 away from 100 (or 5).. like x+100 +1..?

Answer 20

i don't mean to torture you
i will tell you if you like, but then you will say "oh doh"

Answer 21

x+4+1

Answer 22

yes, you would take one away from 101!
so what would you make \(x\) to get \(\sqrt{100}\) in there?

Answer 23

i have \(\sqrt{x+101}\) and i wish it was \(\sqrt{100}\) what does that make \(x\)??

Answer 24

haha.. let me take some time.. i want to be sure im right.. joking, x=10

Answer 25

no not ten

Answer 26

so it's 10sqrt(x+1)

Answer 27

huh?

Answer 28

ok i am done torturing you
if i have \(\sqrt{x+101}\) and i wish it was \(\sqrt{100}\) i would set \(100=x+101\) and solve for \(x\) in one step

Answer 29

oh, i was answering the sqrt(100).. sorry.. is the 10sqrt(x+1 right?

Answer 30

solve \[100=x+101\]

Answer 31

oh.. i see.. x=-1

Answer 32

WHEW!!

Answer 33

haha.. ok i guess im not a lost cause!

Answer 34

now lets check that it is right
\[f(x)=10x+\sqrt{x+101}\]
\[f(-1)=10\times (-1)+\sqrt{-1+101}=-10+10=0\] we got it

Answer 35

no not at all
but you should look for the more or less obvious answer here
ok now that we know
\[f^{-1}(0)=-1\] we can continue
next thing we need is \(f'(x)\)

Answer 36

ohhhh... ok i see.. so with these problems.. you can usually see the relationship between whats under the radical and whats outside of it.. so like 10^2 is 100.. and so i would just find that x=-1.. without the amount of energy it took this time.. nice

Answer 37

if you have not seen this kind of problem before, i can imagine that you would think you have to do something difficult, but usually they make the answer obvious (more or less)
now
\[f(x)=10x+\sqrt{x+101}\]
what is \(f'(x)\) ?

Answer 38

10+(1/2)((x+101)^(-1/2))

Answer 39

10+1/2sqrt(x+101)

Answer 40

ok good, but i don't like it because of the exponential notation
you should memorize the fact that the derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}}\)without resorting to the power rule
it is a very common function, and the derivative never changes
so once you have it memorized, like knowing that \(8\times 9=72\) you can just move on

Answer 41

so we got
\[f'(x)=10+\frac{1}{2\sqrt{x+101}}\] and we also know \(f^{-1}(0)=-1\)
next job is to find
\[f'(-1)\]

Answer 42

yeah i started seeing the relationship the more i did it.. i think i know the answer now.. f'(-1) =10..?

Answer 43

because 10+(1/2*sqrt(100)) = 10+1/20.. sorry i can do math.. 200+1/20

Answer 44

ok good
\[f'(x)=10+\frac{1}{2\sqrt{x+101}}\]
\[f'(-1)=10+\frac{1}{2\sqrt{-1+101}}=10+\frac{1}{20}=\frac{201}{20}\]

Answer 45

and final job is to flip it

Answer 46

since it's 1/f'(f^-1).. so it's 20/201?

Answer 47

aka take the reciprocal
yes

Answer 48

now lets quickly recap, because we spent a lot of time, so lets get exactly what we wanted, and what we needed okay?

Answer 49

when you wrote the equation (different from what I was given) how did you know to make the denominator equal to the inverse(0)? i want to say its because f(x)' or in this case f(inv)prime is related to dx?

Answer 50

okay yeah

Answer 51

here we go
we have
\[f(x)=10x+\sqrt{x+101}\] and you want
\[(f^{-1})'(0)\]
we are going to use
\[(f^{-1})')(0)=\frac{1}{f'(f^{-1}(0))}\] for this we need
\[f^{-1}(0)\] which, after some work, we find \(f^{-1}(0)=-1\)
then we need
\[f'(x)=10+\frac{1}{2\sqrt{x+101}}\] and
\[f'(-1)=\frac{201}{20}\] putting it all together we get
\[(f^{-1})'(0)=\frac{1}{f'(f^{-1}(0))}=\frac{1}{f'(-1)}=\frac{1}{\frac{201}{20}}=\frac{20}{201}\]

Answer 52

Got it.. i just redid it from the top with different numbers and i got them right (online)! Wohoo

Answer 53

Thank you so much! i have an exam coming up so this has been really helpful!

Answer 54

afwan

Answer 55

haha.. not arab but kind gesture.. :)

Answer 56

but you knew what it meant, right?

Answer 57

yeah.. i hang around arabs.. shukran is thank you in both languages (mine is urdu).. and i'm not sure how to say you're welcome in urdu.. need to learn that =/

Answer 58

hmmm if your mother is like mine, she is very disappointed that you do not know her native language
i surely don't
my arabic is limited to please, thank you, and food

Answer 59

the problem with urdu is it's comprised of like a bunch of different languages.. like arabic, hindi, farsi and all of the related ones.. my parents speak different dialects so it was just easier to speak english at home.. but i was brought up learning urdu (till age 4 or 5).. but i really want to learn farsi/arabic.. mom knows farsi since she was born near the boarder

Answer 60

arabic is awesome.. you should learn it by watching tv in arabic!

Answer 61

my father is american, so only english at home
i thought about it, but it is hard, new alphabet, rules etc
i'll stick with the food
good luck on your exam, i am sure you will do well

Answer 62

food is always good.. and thanks! i'll prob be posting more problems soon haha