An inverse function or an anti function is defined as a function which can reverse another function. In simple words, if any function “f” takes x to y then, the inverse of “f” i.e. “f1” will take y to x. Learn more about the inverse of functions in detail from here. The concepts that are covered include:
 What are Inverse Functions?
 How to Find the Inverse of a Function?
 Common Inverse Functions
 Inverse of Trigonometric Functions
 Inverse of Rational Functions
 Inverse of Hyperbolic Functions
 Inverse Logarithmic Functions and Inverse Exponential Functions
 Example Questions from Inverse of Functions
 Inverse of Functions Worksheet
What is an Inverse Function?
A function accepts values, performs particular operations on these values and generates an output. The inverse function accepts the resultant, performs an operation and reaches back to the original function.
If you consider f and g are inverse functions, f(g(x)) = g(f(x)) = x. A function that consists of its inverse functions fetches the original value.
Important Inverse Function Notes:
The inverse of a function – The relation that is developed when the independent variable is interchanged with the variable that is dependent in a specified equation and this inverse may or may not be a function.
Inverse function – If the inverse of a function is a function by itself, then it is known as inverse function, denoted by f1(x).
How to Find the Inverse of a Function?
Generally, the method of calculating an inverse is swapping of coordinates x and y. This newly created inverse is a relation but not necessarily a function.
The original function has to be a onetoone function to assure that its inverse will be also a function. A function is said to be a one to one function only if every second element corresponds to the first value (values of x and y are used only once).
You can apply on the horizontal line test to verify whether a function is a onetoone function. If a horizontal line intersects the original function in a single region, the function is a onetoone function and inverse is also a function.
Common Inverse Functions
The inverses of some of the most common inverse functions are given below. Even the inverse function of trigonometric function, rational function, hyperbolic function and log functions are also discussed in the following points. First, the list of common inverse function is given which include:
Function 
Inverse Function 
Comment 
+ 
– 

* 
/ 
Don’t divide by 0 
1/x 
1/y 
x and y not equal to 0 
x2 
√y 
x and y >= 0 
xn 
y1/n 
n is not equal to 0 
ex 
ln(y) 
Y > 0 
ax 
loga(y) 
y and a > 0 
Sin (x) 
Sin1 (y) 
– π/2 to + π/2 
Cos (x) 
Cos1 (y) 
0 to π 
Tan (x) 
Tan1 (y) 
– π/2 to + π/2 
Inverse of Trigonometric Functions
The inverse trigonometric functions are also known as arc function as they produce the length of the arc which is required to obtain that particular value. There are six inverse trigonometric functions which include arcsine (sin1), arccosine (cos1), arctangent (tan1), arcsecant (sec1), arccosecant (cosec1), and arccotangent (cot1). To learn more about the inverse trigonometric functions along with their graphs, follow the linked article.
Inverse Rational Function
A rational number is a number of form f(x) = P(x)/Q(x) where Q(x) is ≠ 0. To find the inverse of a rational number, follow the following steps. An example is also given below which can help to understand the concept better.
 Step 1: Replace f(x) = y
 Step 2: Interchange x and y
 Step 3: Solve for y in terms of x
 Step 4: Replace y with f1(x) and the inverse function is obtained.
Example:
Find the inverse for the function f(x) = \(\frac{3x+2}{x1}\)
Solution:
First, replace f(x) with y and the function becomes,
y = \(\frac{3x+2}{x1}\)
By replacing x with y we get,
x = \(\frac{3y+2}{y1}\)
Now, solve y in terms of x :
x (y – 1) = 3y + 2
=> xy – x = 3y +2
=> xy – 3y = 2 + x
=> y (x – 3) = 2 + x
=> y = (2 + x) / (x – 3)
So, y = f1(x) = \(\frac{x+2}{x3}\)
Inverse Hyperbolic Functions
Just like the inverse trigonometric function, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are 6 main inverse hyperbolic functions which include sinh1, cosh1, tanh1, csch1, coth1, and sech1. Check out inverse hyperbolic functions formula to learn more about these function in detail.
Inverse Logarithmic Functions and Inverse Exponential Function
The natural log functions are inverse of the exponential functions. Check the following example to understand the inverse exponential function and logarithmic function in detail. Also, get more insights of how to solve similar questions and thus, develop problemsolving abilities.
Example:
Find the inverse of the function f(x) = Ln(x – 2)
Solution:
First, replace f(x) with y
So, y = Ln(x – 2)
Replace the equation in exponential way , x – 2 = ey
Now, solving for x,
x = 2 + ey
Now, replace x with y and thus, f1(x) = y = 2 + ey
Example Question from Inverse Function
To solve for an equation:
f(x) = 2x + 3, at x = 4
We have,
f(4) = 2 x 4 + 3
f(4) = 11
Now, let’s apply for reverse on 11.
f1(11) = (11 – 3) / 2
f1(11) = 4
Magically we get 4 again
Therefore, f1(f(4)) = f(4)
So, when we apply function f and its reverse f1 gives the original value back again, i.e, f1(f(x)) = x
An example illustrating inverse functions 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: 
y3 
= 
2x 
Divide both sides by 2: 
(y3)/2 
= 
x 
Swap sides: 
x 
= 
(y3)/2 
Solution (put “f1(y)” for “x”) : 
f1(y) 
= 
(y3)/2 
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