In Exercises 35-38 : a. Find f^-1(x) b. Graph f and f^-1...

In Exercises 35-38 :
a. Find $f^{-1}(x)$
b. Graph $f$ and $f^{-1}$ together.
c. Evaluate $d f / d x$ at $x=a$ and $d f^{-1} / d x$ at $x=f(a)$ to show that at these points $d f^{-1} / d x=1 /(d f / d x)$
$$f(x)=2 x^{2}, \quad x \geq 0, \quad a=5$$

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Key Concepts

Inverse Functions

An inverse function reverses the operation of a given function. When a function f and its inverse f?¹ are composed, they return the original input (i.e., f(f?¹(x)) = x and f?¹(f(x)) = x). Understanding inverses is fundamental in algebra and calculus, especially when solving equations and analyzing function behavior.

Graphing Functions and Their Inverses

Graphing a function together with its inverse reveals a symmetry about the line y = x. This graphical property provides a visual confirmation of the inverse relationship and helps in understanding the domain and range as swapped values between the function and its inverse.

Derivative of an Inverse Function

The derivative of an inverse function at a given point is given by the reciprocal of the derivative of the original function evaluated at the corresponding point, expressed as (f?¹)'(x) = 1 / (f'(f?¹(x))). This relationship is crucial in calculus for understanding how the rates of change of functions and their inverses are interrelated.

Chain Rule

The chain rule is a method for finding the derivative of a composite function. It plays a key role in deriving the formula for the derivative of an inverse function by linking the derivative of the composite function to the derivatives of its individual functions. This concept is central to differential calculus.

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