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 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 $g(y)= x$.
  • Inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then putting $y$ into the inverse function $g$ produces the output $x$, and vice versa. i.e., $f(x)=y$, and $g(y)=x$.
  • 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.
  • A function $f$ and its inverse $f^{-1}$.

• Inverse Functions

  • An inverse function is a function that undoes another function.
  • A function $f$ that has an inverse is called invertible; the inverse function is then uniquely determined by $f$ and is denoted by $f^{-1}$.
  • 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.
  • A function $f$ and its inverse, $f^{-1}$.

• 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.

• 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, 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.