inversion

Examples of inversion in the following topics:

• The Existence of Inverse Functions and the Horizontal Line Test

  • Recognize whether a function has an inverse by using the horizontal line test

• Restricting Domains to Find Inverses

  • Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
  • Below is the graph of the parabola and it's "inverse."  
  • Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.
  • Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.  
  • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

• Inverse Functions

  • An inverse function is a function that undoes another function.
  • Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.
  • Not all functions have an inverse.
  • Thus, the inverse of $x^2+2$ is $\sqrt{x-2}$.
  • A function $f$ and its inverse, $f^{-1}$.

• Inverse Trigonometric Functions

  • In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
  • Recall that the inverse of sine is arcsine (denoted arcsin), the inverse of cosine is arccosine (denoted arccos), and the inverse of tangent is arctangent (denoted arctan).
  • Inverse functions are denoted by an exponent of -1.
  • The inverse sine function is notated $\arcsin x$.
  • The inverse tangent function notated $\arctan x$.

• Introduction to Inverse Functions

  • To find the inverse function, switch the $x$ and $y$ values, and then solve for $y$.
  • An inverse function, which is notated $f^-1(x) $, is defined as the inverse function of $f(x)$ if it consistently reverses the $f(x)$ process.
  • More concisely and formally, $f^-1(x)$ is the inverse function of $f(x)$ if:
  • In general, given a function, how do you find its inverse function?
  • Remember that an inverse function reverses the inputs and outputs.  

• Inverse Functions

  • An inverse function is a function that undoes another function: For a function $f(x)=y$ the inverse function, if it exists, is given as $g(y)= x$.
  • A function $f$ that has an inverse is called invertible; the inverse function is then uniquely determined by $f$ and is denoted by $f^{-1}$ (read f inverse, not to be confused with exponentiation).
  • Not all functions have an inverse.
  • Inverse operations are the opposite of direct variation functions.
  • A function $f$ and its inverse $f^{-1}$.

• Chromosomal Structural Rearrangements

  • A pericentric inversion that is asymmetric about the centromere can change the relative lengths of the chromosome arms, making these inversions easily identifiable.
  • When one homologous chromosome undergoes an inversion, but the other does not, the individual is described as an inversion heterozygote .
  • This inversion is not present in our closest genetic relatives, the chimpanzees.
  • Pericentric inversions include the centromere, and paracentric inversions do not.
  • A pericentric inversion can change the relative lengths of the chromosome arms; a paracentric inversion cannot.

• The Inverse of a Matrix

  • Having seen that the number 1 plays a special role in multiplication, because $1x=x$, the inverse of a number is defined as the number that multiplies by that number to give 1. b is the inverse of a if the inverse of a matrix multiplies by that matrix to give the identity matrix.
  • Note also that only square matrices can have an inverse.
  • Find the inverse of$\begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix}$.
  • However, in some cases, the inverse of a square matrix does not exist.
  • Practice finding the inverse of a matrix and describe its properties

• Inverse Trigonometric Functions: Differentiation and Integration

  • It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.
  • The inverse trigonometric functions are also known as the "arc functions".
  • There are three common notations for inverse trigonometric functions.
  • They can be thought of as the inverses of the corresponding trigonometric functions.
  • The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions.

• Finding Angles From Ratios: Inverse Trigonometric Functions

  • The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.
  • In order to solve for the missing acute angle, use the same three trigonometric functions; however, use the inverse key ($^{-1}$on the calculator) to solve for the angle      ($A$) when given two sides.
  • (Soh from SohCahToa)  Write the equation and solve using the inverse key for sine.
  • Recognize the role of inverse trigonometric functions in solving problems about right triangles