You are here:

Calculus/Inverses of functions.

Advertisement

Expert: - 3/29/2008


Question
Let f be the function defined by f(x)=x^3 + x. If
g(x)= f^-1(x)and g(2)=1, what is the value of g'(2)?

I thought it may be done similar to composites but I am unsure of how to begin.

Answer

Inverses.
Questioner:   Allie
Category:  Calculus
Private:  No
 
Subject:  implicit differentiation
Question:  Let f be the function defined by f(x)=x^3 + x. If
g(x)= f^-1(x)and g(2)=1, what is the value of g'(2)?

I thought it may be done similar to composites but I am unsure of how to begin.
...............................
Hi, Allie,

If  f(x) = x^3 + x, and g(x) is its inverse, which exists, btw, then of course  g(2) = 1, because f(1) = 2. [That's what it means.]

Now the value of g'(2) can be obtained this way: (You seemed to know it had something to do with implicit differentiation.

y = x^3 + x
1 = (3x^2 + 1) dx/dy

dx/dy = 1/(3x^2 + 1), at  x = 1.  << NOT  x = 2.

dx/dy = 1/(3 + 1) = 1/4

See attached graph. The points to look at are (1,2) on the original and (2,1) on the inverse  y = g(x).

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

©2012 About.com, a part of The New York Times Company. All rights reserved.