Know About Inverse Functions and its Domain and Range

Table of Contents

A function that can be reversed into another function is known as an inverse function, sometimes known as an anti function. If a function “f” goes from x to y, its inverse will go from y to x. If the function is indicated by ‘f’ or ‘F,’ the inverse function is denoted by f^-1 or F^-1. (-1) should not be confused with exponent or reciprocal here.

If f and g are Inverse Functions, then f(x) = y only if and only if g(y) = x.

The inverse sine function is used in trigonometry to get the measure of the angle for which the sine function generated the value. For instance, sin^-1(1) = sin^-1(sin 90) = 90°. As a result, sin 90° equals 1.

Let us learn more about the Inverse Function, its types, the steps to find the inverse function, and the Domain and Range of a Function like Inverse Function.

Inverse Function

A function takes values, executes specific operations on them, and produces an output. The inverse function agrees with the resultant, runs, and returns to the original function.

The inverse function returns the initial value for which the output of a function was given.

When it comes to functions, f and g are inverse, with f(g(x)) = g(f(x)) = x. A function that is composed of its inverse returns the original value.

For example: f(x) = 2x + 5 = y

The inverse of f is hence g(y) = (y-5)/2 = x. (x).

Types of Inverse Function

Inverse functions include the inverse of trigonometric functions, rational functions, hyperbolic functions, and log functions. Let’s look at some of the most important inverse functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are sometimes known as arc functions because they calculate the length of the arc required to obtain the desired result. The six inverse trigonometric functions are arcsine (sin^-1), arccosine (cos^-1), arctangent (tan^-1), arcsecant (sec^-1), arccosecant (cosec^-1), and arccotangent (cot^-1).

Inverse Rational Function

A rational function is one that has the form f(x) = P(x)/Q(x), where Q(x) equals zero.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of hyperbolic functions, just like inverse trigonometric functions. Inverse hyperbolic functions include sinh^-1, cosh^-1, tanh^-1, coth^-1, Cosech^-1 and sech^-1.

Inverse Logarithmic Functions and Inverse Exponential Function

The natural log functions are the inverse of the exponential functions.

Read Also: Utilization of Complex Number and its Characteristics

Steps To Find An Inverse Function

The steps below will assist you in determining the inverse of a function fast. In this case, we take the function f(x) = ax+b and try to find its inverse.

To get y = ax+ b, replace f(x) = y with the given function f(x) = ax + b.
To get x = ay + b, substitute x with y and y with x in the method y = ax + b.
Now solve the x = ay + b for y. we get, y = (x – b/a).
Finally, we can replace y = f^-1(x) by f^-1(x) = (x – b)/a.

Domain and Range of the Inverse Function

To find the algebraic formula for the inverse of a function y=f(x), first, switch the roles of y and x to get x=f(y), then solve this statement for y to get y=f1 (x). By switching the roles of y and x, the domain and range of f are effectively interchanged (x).

Domain and Range of an Inverse Function

Assume we have a function f(x) and its inverse is f−1(x). The range of f(x) is then the domain of the inverse function f−1(x), and the domain of f−1(x) is the domain of f. (x).