Inverse Functions

About Inverse Functions

The function that can reverse into another function is known as an inverse function or anti-function. Simply put, if a function "f" converts x to y, its opposite will convert y to x. The inverse function is denoted by f-1 or F-1 if the function is denoted by 'f' or 'F'. (-1) must not be confused with exponent or reciprocal.
f(x) = y if and only if g(y) = x if f and g are inverse functions.
The inverse sine function is used in trigonometry to determine the measure of angle for which the sine function generated the value. Sin-1(1), for example, equals sin-1(sin 90) = 90° . As a result, sin 90° equals 1.

Definition of Inverse Functions

  • A function takes input and performs specific actions on it before returning a result. The inverse function agrees with the resultant, performs its purpose, and returns to the original function.
  • The inverse function returns the original value of the result of a function.
  • When it comes to functions, the inverse of f and g is f(g(x)) = g(f(x)) = x. The original value is returned by a function that is its inverse.
  • Note: When the independent variable is replaced with a variable that is dependent on a given equation, the inverse is formed, which may or may not be a function.
  • The inverse function, indicated by f-1, is when the inverse of a function is itself (x).

Types of Inverse Function

  • Inverse functions include the inverse of trigonometric functions, rational functions, hyperbolic functions, and log functions, among others. Below are the inverses of some of the most frequent functions.
Function The inverse of the Function Comment
+ -  
× / Don’t divide by 0
1/x 1/y x and y are not equal to 0
x2 √y x and y ≥ 0
xn y1/n n is not equal to 0
ex ln(y) y > 0
axe log a(y) y and a > 0
Sin (x) Sin-1(y) - π/2 to + π/2
Cos(x) Cos-1(y) 0 to π
Tan(x) Tan-1(y) - π/2 to + π/2

Inverse Trigonometric Functions
Inverse trigonometric functions are sometimes known as arc functions since they yield the length of the arc needed to acquire a given number. Arcsine (sin-1), arccosine (cos-1), arctangent (tan-1), arcsecant (sec-1), arccosecant (cosec-1), and arccotangent (cot-1) are the six inverse trigonometric functions.

Inverse Rational Function
A rational function has the form f(x) = P(x)/Q(x), where Q(x) is less than 0. Follow these procedures to obtain the inverse of a rational function.

  1. Change f(x) to y.
  2. Swap the x and y coordinates.
  3. Find y in terms of x.
  4. Substitute f-1(x) for y to get the inverse of the function.
Finding Inverse Function Using Algebra. Put “y” for “f(x)” and solve for x:
The function: f(x) 2x + 3
Put “y” for “f(x)”: y 2x + 3
Subtract 3 from both sides: y - 3 2x
Divide both sides by 2: (y-3)/2 x
Swap sides: x (y - 3)/2
Solution (put “f-1(y)” for “x”) : f - 1(y) (y - 3)/2

 

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Inverse Functions
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