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.
- Change f(x) to y.
- Swap the x and y coordinates.
- Find y in terms of x.
- Substitute f-1(x) for y to get the inverse of the function.
| 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 |
Get a List of all Maths Formulas on one page.
Download the pdf of Inverse Functions
