Implicit Functions, the Implicit Function Theorem, and Implicit Differentiation

All flashcards

What is an implicit equation?

It is a relation among variables written as $R(x_1,\dots,x_n)=0$, rather than necessarily solving explicitly for one variable.

What is an implicit function?

A function specified by an equation relating its value to its inputs, even when that equation has not been rearranged into an explicit formula.

How is an inverse function defined implicitly?

If $y=g(x)$ and $g$ has a unique inverse, solving for $x$ gives $x=g^{-1}(y)$. The equation identifies the inverse's value from the input $y$.

Why can the circle relation yield a multivalued function?

For most $x$ with $-1<x<1$, the equation $x^2+y^2=1$ gives two values, $y=\sqrt{1-x^2}$ and $y=-\sqrt{1-x^2}$.

How can a multibranch relation sometimes be made into a single-valued function?

Restrict the domain or select one branch of the graph, such as using only the upper half of the unit circle.

Quiz questions

1. What is the scope of the implicit function theorem's guarantee near a point where its conditions hold?Answer: It guarantees a differentiable representation in a sufficiently small neighborhood of that point.
2. If $g(x)=2x-1$, what is its inverse function?Answer: $g^{-1}(y)=(y+1)/2$
3. What can make an algebraic relation multivalued as a function of its input?Answer: The defining polynomial can have multiple output solutions for the same input.
4. What does the relation $x=0$ show when considering $y$ as a function of $x$?Answer: It describes a vertical line and does not give a unique $y$ for its input value.
5. For a general polynomial equation of degree five or higher in $y$, which statement is accurate?Answer: A solution in familiar closed form is not generally available.