Function within a function problem

• Apr 26th 2009, 03:00 PM
mamori
Function within a function problem
Hello again! I am again lost on another problem. I wasn't sure what to call this, so hopefully it's still somewhat understandable. Thank you very much in advance!

The functions f and g are differentiable. For all x, f(g(x)) = x and g(f(x)) = x. If f(3) = 8 and f'(3) = 9, what are the values of g(8) and g'(8)?

(A) g(8) = 1/3 and g'(8) = -1/9
(B) g(8) = 1/3 and g'(8) = 1/9
(C) g(8) = 3 and g'(8) = -9
(D) g(8) = 3 and g'(8) = -1/9
(E) g(8) = 3 and g'(8) = 1/9
• Apr 26th 2009, 03:15 PM
derfleurer
f(g(x)) = x and g(f(x)) = x

This means that these functions are inverses.

Can you take it from there?
• Apr 26th 2009, 03:15 PM
Jester
Quote:

Originally Posted by mamori
Hello again! I am again lost on another problem. I wasn't sure what to call this, so hopefully it's still somewhat understandable. Thank you very much in advance!

The functions f and g are differentiable. For all x, f(g(x)) = x and g(f(x)) = x. If f(3) = 8 and f'(3) = 9, what are the values of g(8) and g'(8)?

(A) g(8) = 1/3 and g'(8) = -1/9
(B) g(8) = 1/3 and g'(8) = 1/9
(C) g(8) = 3 and g'(8) = -9
(D) g(8) = 3 and g'(8) = -1/9
(E) g(8) = 3 and g'(8) = 1/9

First from $\displaystyle g(f(x)) = x$ sub. x=3. Next differentiate $\displaystyle g(f(x)) = x$ wrt x so $\displaystyle g'(f(x))\cdot f'(x) = 1$, then subs x= 3 again.
• Apr 26th 2009, 03:17 PM
derfleurer
You're thinking way too much. =p
• Apr 26th 2009, 03:19 PM
Calculus26
1.

since g(f(x)) = x

g(f(3)) = 3

But f(3) = 8

so g(8) = 3

2. f(g(x)) = x

f and g are inverses so you have a theorem which tells you

g ' (8) = 1/ f'(3)

In general g ' (c) = 1/ f ' (a) where f(a) = c

Or From first principles:

[f(g(x)] ' = 1 for all x

But [f(g(x)] ' also equals f ' (g(x)) g ' (x)

at x = 3 f ' (g(8)) g ' (8) = 1

f ' (3) *g ' (8) =1

9* g ' (8) =1

g' (8) =1/9

f and g are inverses so you have a theorem which tells you

g ' (8) = 1/ f'(3)
• Apr 26th 2009, 03:23 PM
Jester
Quote:

Originally Posted by derfleurer
You're thinking way too much. =p

Quote:

Originally Posted by Calculus26
1.

since g(f(x)) = x

g(f(3)) = 3

But f(3) = 8

so g(8) = 3

2. f(g(x)) = x

f and g are inverses so you have a theorem which tells you

g ' (8) = 1/ f'(3)

In general g ' (c) = 1/ f ' (a) where f(a) = c

Or From first principles:

[f(g(x)] ' = 1 for all x

But [f(g(x)] ' also equals f ' (g(x)) g ' (x)

at x = 3 f ' (g(8)) g ' (8) = 1

f ' (3) *g ' (8) =1

9* g ' (8) =1

g' (8) =1/9

f and g are inverses so you have a theorem which tells you

g ' (8) = 1/ f'(3)

Apparently me and Calc26 (Rofl)
• Apr 26th 2009, 03:34 PM
mamori
Hi everyone! Thank you so much for all your help! I did understand that the functions are inverses, but I forgot the theorem for the derivatives. So thanks for your explanation, Calculus26! I do have one question. Where does

But [f(g(x)] ' also equals f ' (g(x)) g ' (x)

come from? Another theorem? I never learned this theorem/rule, so it's a bit tough on understanding this.
• Apr 26th 2009, 03:39 PM
derfleurer
That's just chain rule.

sin(x^2)' = cos(x^2)(2x)

And if you understood that they were inverses, you didn't really need to remember any calculus rules. Just know that the slope of the inverse is 1/m and that your domain/range swap. So f(3) = 8 yields f^-1(8) = 3.
• Apr 26th 2009, 03:45 PM
Calculus26
Danny-- I thought we were paid by the number of words
• Apr 26th 2009, 03:46 PM
mamori
Quote:

Originally Posted by derfleurer
That's just chain rule.

sin(x^2)' = cos(x^2)(2x)

And if you understood that they were inverses, you didn't really need to remember any calculus rules. Just know that the slope of the inverse is 1/m and that your domain/range swap. So f(3) = 8 yields f^-1(8) = 3.

Thanks. =] I just understood everything on this page. Somehow that inverse slope rule slipped my mind. Thanks a million everyone!
• Apr 26th 2009, 03:49 PM
Jester
Quote:

Originally Posted by Calculus26
Danny-- I thought we were paid by the number of words

Don't we wish (Bigsmile)