What is an inverse function?

Mathematics is a subject with various disciplines and subtopics. Arithmetics, geometry, exponential, percentage are some of these branches of mathematics that deal with the calculation, analysis, and manipulation of numbers as well as symbols. Algebra is also one of the disciplines of mathematics that deals with the manipulation of symbols which represent the unknown value in the equation. The equation of algebra is consists of coefficients, variables, and constants.

Function

The function is a mathematical expression or a relation between the dependent and independent variables. Functions are generally represented f(x). Other than the symbols f(x) use of g(x) or P(x) is also seen in some relations to represent a function. And, the relation between two sets of variables is represented by 

y = f(x)

For that relation, y = f(x), for every x, there will be a unique value of y. The relation will give only one value at a time as an output.

Types of algebraic functions

Functions are divided into different types on the basis of variables and their way of representation. These algebraic functions are described below,

• Linear Function: Linear function includes one dependent and one independent variable in the expression. The expression is written in the form of y = mx + c
• Quadratic Function: Quadratic function includes a 2 degree polynomial in its expression. The expression is written in the form of g(x) = ax² + bx + c.
• Cubic Function: Cubic function includes a 3 degree polynomial in its expression. The expression is written in the form of g(x) = ax³ + bx² + cx + d.
• Polynomial Function: A polynomial function includes an n-degrees polynomial in the expression. The expression is written in the form of g(x) = c_nxⁿ + c_{n – 1}x^{n – 1} + … + c₂x² + c₁x + c₀.
• Rational Function: Rational function is written in the form of g(x) = p(x)/q(x), where p(x) and q(x) stands for polynomial functions.
• Radical Function: Radical function is written in the form of g(x) = , where q(x) is a polynomial function.

What is an inverse function?

Answer:

An inverse function or also widely known as “anti function” is a function that reverses the result of given another function.Such as if an f(x) = 11, then, its inverse function will be f^-1(x) = -11.

The inverse functions generally used  for common functions,

Function	Inverse	Descriptions
+	–	(+) changes to (-)
x	÷	Should not be divided by 0
1/x	1/y	x and y, not zero.
x2	√y	x and y=0
xn	1/yn	n not to be zero
ex	In(y)	y>0
ax	loga(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

Types of Inverse functions:

There are different types of inverse functions. They are Inverse trigonometric functions, Inverse rational functions, and  Inverse hyperbolic functions. Let’s take a look at these in more detail,

• Inverse Trigonometric Function

Inverse functions are responsible to give the length of the arc which is required to attain a particular value. The inverse trigonometric function is also widely known as the arc function. There are basically six inverse trigonometric functions.

1. arcsine (sin^-1)
2. arccosine (cos^-1)
3. arctangent (tan^-1)
4. arcsecant (sec^-1)
5. arccotangent (cot^-1)
6. arccosecant (cosec^-1)

• Inverse Rational Function

It is the function represented in the form of f(x) = P(x)/Q(x)

Where Q(x) ≠ 0

• Inverse Hyperbolic Function

Inverse hyperbolic functions are the inverse of hyperbolic functions. The six types of inverse hyperbolic functions are sinh^-1, cosh^-1, tanh^-1, coth^-1, sech^-1, cosech^-1

Sample Problems

Question 1: Find the inverse function of f(x) = y = 3x + 2/x – 1

Solution:

f(x) = y = 3x + 2/x – 1

y(x – 1) = 3x + 2

yx – y = 3x + 2

yx = 3x = y + 2

x(y – 3) = y + 2

x = f^-1(y) = y + 2/y – 3

Question 2: check the function f(x) = 5x – 2 if, x = 4. and find the inverse function.

Solution:

f(x) = 5x – 2

f(4) = 5 × 4 – 2

f(4) = 18

Then,

f^-1(x) = (18 + 2)/5

f^-1(x) = 20/5

f^-1(x) = 4

Question 3: Find the inverse of function f(x) = -2x + 7/x

Solution:

f(x) = -2x + 7/x

y = -2x + 7/x

x = -2y + 7/y

xy = -2y + 7

xy + 2y = 7

y(x + 2) = 7

y = 7/x + 2

f^-1(x) = 7/x + 2