Question

verify the inverse using definition of inverse

Answer 1

\[f(x)=x^2-10x\]
\[f ^{-1}(x)=5+\sqrt{x+25}\]

Answer 2

the definition of inverse says,
\(f(f^{-1}) (x) = x\)
or
\(f^{-1}f(x)=x\)
verify any 1 of these.

Answer 3

?? ok so
\[f^-1(x^2-10x)\]
\[5+\sqrt{x^2-10x+25}\] ???

Answer 4

factor x^2-10x+25
hint : its a perfect square trinomial
can you ?

Answer 5

(x-5)(x-5)

Answer 6

correct!
\((x-5)(x-5)=(x-5)^2 \\ \sqrt{(x-5)^2}=x-5\)
can you finish it ?

Answer 7

dont i have to square both sides to get rid of square root ?

Answer 8

you can get rid of the square root just by using \(\sqrt a^2=a\)

Answer 9

ok?

Answer 10

\(5+\sqrt{x^2-10x+25}= 5 + (x-5)\)
do you get that = x ?
if yes, they both are inverse of each other :)

Answer 11

ok so it is inverse

Answer 12

yes, thats what you needed to verify.

Answer 13

so now dont i need to do \[f(f^-1(x))\]
to see if they match

Answer 14

@hartnn

Answer 15

please do it, to be absolutely sure about it.

Answer 16

so
\[(5+\sqrt{x+25})^2-10(5+\sqrt{x+25})\]

Answer 17

yes, just use the general formula
\((a+b)^2=a^2+2ab+b^2\)

Answer 18

??

Answer 19

i am just hating these square roots thats all its confusing me

Answer 20

\((5+\sqrt{x+25})^2-10(5+\sqrt{x+25}) \\= (25+2\times 5\times \sqrt{x+5}+(\sqrt{x+5}^2))-10(5+\sqrt{x+25}) \\ = ....?\)

its not difficult, lenghty yes, but you need to be patient :)

Answer 21

sorry, i meant 25 instead of 5 inside square root

Answer 22

so i dont know does it work out to be x doesnt seem like it for some reason

Answer 23

i will work out nicely....
\((5+\sqrt{x+25})^2-10(5+\sqrt{x+25}) \\= (25+2\times 5\times \sqrt{x+25}+(\sqrt{x+25}^2))-10(5+\sqrt{x+25}) \\ = 25 +10\sqrt{x+25}+x+25-50-10\sqrt{x+25}\)

now its easy to simplify...the huge term gets cancelled...

Answer 24

so not inverse functions?

Answer 25

oops never mind

Answer 26

you will get 'x' and they are inverses :)

Answer 27

thank you so much @hartnn for your patience

Answer 28

you're most welcome ^_^

Answer 29

so one more quick question - since x^2-10x and my other function these are then not one to one are they

Answer 30

:)

Answer 31

???

Answer 32

ok, why not ?
they are one to one functions,
one value of 'x' fives you only one value of f(x) and f^-1 x

Answer 33

**gives

Answer 34

\[f(x)=x^2-10x, x \ge 5\]
so how do u graph this as x^2-10x is a parabola and our other one is not

Answer 35

http://awesomescreenshot.com/07d1unslb1

Answer 36

so you want to graph the inverse function ?

Answer 37

:) yes

Answer 38

those graphs are correct, whats the doubt ?

Answer 39

well i guess just dont understand how as in our directions says graph the inverse function (create chart switching x and y reflect the graph of f across diagonal line and then verify the the inverse function formula matches the graph) guess just not seeing that how about the diagonal line

Answer 40

i guess i dont see how it reflects

Answer 41

You aren't suppose to have part of that parabola there since your restricted it's domain so the function be one to one and thus having an inverse?

Answer 42

thanks for all the help appreciate it :D