Question

Find the Inverse

Answer 1

\(\large a = f^{-1}(c) \implies f(a) = c\)

Answer 2

You're given :
\(\large f(x) = 4x+6x^7 ; c = -10\)

Answer 3

\(\large f(a) = c\)

\(\large 4a + 6a^7 = -10\)

solve \(\large a\)

Answer 4

By inspection \(\large a = -1\)

Answer 5

yeah i see that now.

Answer 6

good, next find the derivative

Answer 7

of 4a + 6a^7?

Answer 8

4 + 56a^6

Answer 9

\[\large f^{-1}(-10) = \dfrac{1}{f'(-1)}\]

Answer 10

42, sorry, lol

Answer 11

\(\large f(x) = 4x + 6x^7\)
\(\large f'(x) = ?\)
\(\large f'(-1) = ?\)

Answer 12

f'(x) = 4 + 42x^6
F'(x) = 46

Answer 13

yes, plug that into the formula

Answer 14

1/46

Answer 15

Yep !

Answer 16

heyyy, it worked, thank you

Answer 17

ok so recap, set f(a) to c, find a, derive f(a) and set it to the -1 power

Answer 18

and the second one i got 9 and 1/5 :)

Answer 19

yes if that makes it easy to remember :)

Answer 20

in general :

\[\large \left(f^{-1}(f(x))\right)' = \dfrac{1}{f'(x)}\]

Answer 21

i see so the inverse of any function is just the derivative under 1

Answer 22

the `derivative of inverse of a function at x=c` is the `reciprocal of derivative of the original function at ` \(x = f^{-1}(c)\)

Answer 23

i see, so c is the point at which the graph is.. inversed""

Answer 24

\[\large \left(f^{-1}(c)\right)' = \dfrac{1}{f'(f^{-1}(c))}\]

Answer 25

yes, (c, f(c)) is a point on original function
so (f(c), c) will be a point on its inverse function

Answer 26

but its not necessarily the vertex of where the graph is being flipped?

Answer 27

can you sketch the inverse of f(x) ?