inverse function

Examples of inverse function 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

• Introduction to Inverse Functions

  • 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.
  • Below is a mapping of function $f(x)$  and its inverse function, $f^{-1}(x)$.  
  • In general, given a function, how do you find its inverse function?
  • Since the function $f(x)=x^2$ has multiple outputs, its inverse is not a function.
  • A function's inverse may not always be a function.  

• Inverse Trigonometric Functions

  • Each trigonometric function has an inverse function that can be graphed.
  • 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.
  • An exponent of $-1$ is used to indicate an inverse function.
  • The inverse sine function can also be written $\arcsin x$.
  • The inverse cosine function can also be written $\arccos x$.

• 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.
  • More concisely and formally, $f^{-1}x$ is the inverse function of $f(x)$ if $f({f}^{-1}(x))=x$.
  • 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.
  • This function fails the horizontal line test, and therefore does not have an inverse.

• 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, but, use the inverse key ($^{-1}$on the calculator) to solve for the angle ($A$) when given two sides.
  • Therefore, use the sine trigonometric function.
  • (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

• Introduction to Exponential and Logarithmic Functions

  • Logarithmic functions and exponential functions are inverses of each other.
  • The inverse of an exponential function is a logarithmic function and vice versa.
  • In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue.
  • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.
  • As they are inverses composing these two functions in either order yields the original input.

• Inverses of Composite Functions

  • A composite function represents, in one function, the results of an entire chain of dependent functions.
  • In mathematics, function composition is the application of one function to the results of another.
  • A composite function represents in one function the results of an entire chain of dependent functions.
  • This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if $Y$ coincides with the co-domain of $f$.
  • Let's go through the relationship between inverses and composition in this example. let's take two functions, compose and invert them.

• Graphs of Logarithmic Functions

  • When graphing without a calculator, we use the fact that the inverse of a logarithmic function is an exponential function.
  • Logarithmic functions can be graphed by hand without the use of a calculator if we use the fact that they are inverses of exponential functions.  
  • Now let us consider the inverse of this function.
  • Now we must note that these points are not on the original function ($y=log{_3}x$) but rather on its inverse $3^x=y$.
  • The graph of the logarithmic function with base $3$ can be generated using the function's inverse.

• Direct and Inverse Variation

  • Two variables in direct variation have a linear relationship, while variables in inverse variation do not.
  • Inverse variation is the opposite of direct variation.
  • As an example, the time taken for a journey is inversely proportional to the speed of travel.
  • Thus, an inverse relationship cannot be represented by a line with constant slope.
  • Relate the concept of slope to the concepts of direct and inverse variation

• Solving Systems of Equations Using Matrix Inverses

  • A system of equations can be readily solved using the concept of the inverse matrix and matrix multiplication.
  • A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication.  
  • Using the matrix function on the calculator, first enter both matrices.  
  • Then calculate $[A^{-1}][B]$, that is, the inverse of matrix $[A]$, multiplied by matrix $[B]$.
  • Practice using inverse matrices to solve a system of linear equations