How to Find the Inverse of a Function

In math, finding the inverse of a function means undoing what the original function does. The inverse switches the input and output of the original function. But not all functions have inverses; there are rules to for an inverse to exist for a function.

This article looks at the steps to find a function's inverse and the conditions it must meet.

Understanding Functions

In mathematics, a function is a rule that assigns one unique value from a set to each element in another set (referred to as the domain). Simply put, a function takes an input and produces a specific output.

Definition of a function

The relationship or correspondence that gives each element x in set A precisely one element y in set B is called a function f from set A to set B. This may be expressed as y = f(x), where x is the function's input (or independent variable) and y is its output (or dependent variable).

Important details about functions:

• Domain: Domain is the collection of input values for a function where it is defined and operates.

• Codomain: The range of a function is the set of all possible output values it can produce.

• Uniqueness: Each domain element pairs with one codomain element uniquely.

• Notation: Functions are represented by letters like f, g, or h, with the notation being y = f(x), showing output (y) and input (x).

• Mapping: A function maps domain items to codomain elements, serving as a model.

Domain

• The domain of a function is the collection of all possible input values it can take, represented by Dom(f) or {x | condition}.

• It defines the range of values over which the function is defined.

Range

• The range of a function is the collection of all potential output values it can generate within its domain.

• This range can be denoted as "Range(f)" in mathematical notation or as a set of values {y | condition}, where conditions may restrict the output values.

What is an Inverse Function?

An inverse function "undoes" the operation of another function by mapping outputs back to the original inputs. If function ? maps elements from set ? to set ?, its inverse function, represented as ? ⁻¹, maps items from set ? back to set ?.

The inverse function ensures that ? ⁻¹ ( ? ( ? ) ) = ? for all ? in the domain of ?, and ? ( ? ⁻¹ ( ? ) ) = ? for all ? in the range of ?.

This relationship allows for the reversal of the original function's mapping process. In simpler terms, if f(a) = b, then f^-1(b) = a, where a is the input and b is the output.

Key points about inverse functions:

• Inverse functions are not always possible unless the original function is one-to-one.

• The range of the inverse function is the domain of the original function, and vice versa.

• The symbol ? ⁻¹ is used to represent the inverse of a function f. It should not be confused with exponentiation, where the reciprocal of ? is represented as ? ⁻¹.

Conditions for existence

• One-to-One (Injective): The original function must be one-to-one, meaning each input in the domain corresponds to a unique output in the range. Mathematically, this condition can be expressed as: f(x₁) = f(x₂) ⟹ x , In simpler terms, if f(a) = f(b), then a = b.

• Onto (Surjective): The original feature must be surjective, meaning every element in the range has a corresponding element in the domain. The function covers the entire area without leaving any gaps. Mathematically, this condition can be expressed as: ∀y ∈ Range(f), ∃x ∈ Domain (f) such that f(x) = y. In simpler terms, for every y in the range of f, there exists an x in the domain of f such that f(x) = y.

• Strictly Increasing or Decreasing: A function must be strictly increasing or decreasing throughout its domain to allow for a unique reversal process without overlapping.

• Existence of the Inverse Operation: Functions must be reversible, meaning the operations they perform can be undone in a single step. Addition/subtraction and multiplication/division are examples of inverse operations.

Finding the Inverse of a Function Algebraically

Step-by-step method:

1. Start with the Original Function: First, write down the original function given a function f(x).

2. Replace f(x) with y: Rewrite function equation with y as a variable instead of f(x) in terms of .i.e. Replace f(x) with ? .

3. Swap x and y: Switch x and y in equation to find inverse function representation.

4. Solve for y: Rearrange the equation to solve for y to find the inverse function in terms of y.

5. Replace ? with ?⁻¹(x): Express inverse function y as ?⁻¹(x) in terms of x. i.e. Replace y with ?^−1(x) to express the inverse function explicitly in terms of x.

Solving for x:

To explicitly express the inverse function, you might need to solve for x after determining the inverse function's equation. In order to do this, you must isolate x on one side of the equation.

Finding the Domain and Range of the Inverse Function:

Domain of the Inverse: The original function's range will coincide with the domain of the inverse function.

Range of the Inverse: The domain of the original function will coincide with the range of the inverse function.

Understand by an Example

Find the inverse of the function: f(x) = 2x + 3

Solution:

Step-by-Step Method:

1. Start with the Original Function: Original function: f(x) = 2x + 3

2. Replace f(x) with y: Equation: y = 2x + 3

3. Swap x and y: Swapping x and y gives: x = 2y + 3

4. Solve for y:

• x=2y+3
• 2y=x-3
• y=(x-3)/2

5. Replace y with ?⁻¹(x) :

• Inverse function: ?−1(x) = (x-3)/2

Solving for x:

The inverse function ?−1(x) = (x-3)/2 is already solved explicitly for y, so there's no need to solve for x separately.

Finding the Domain and Range of the Inverse Function:

The domain of the inverse function consists of all real numbers, similar to the range of the original function.

The range of the inverse function includes all real numbers, mirroring the domain of the original function.

Therefore, Inverse function f^-1(x) = (x-3)/2 has domain & range of all real numbers.

Identifying Inverse Functions From a Graph

If we are provided with the graphs of two functions, we can determine if they are inverse functions. If the functions' graphs are symmetrical in relation to the line y=x, we call the functions inverses of one another. This is due to the fact that if a point (x, y) is on the function, then the point (y, x) is on its inverse function.

Solved Examples on Inverse of a Function

Question 1. Finding the Inverse of f(x) = 2x + 1:

Solution:

step 1: Start with the original function: y = 2x + 1.

step 2 : Swap x and y: x = 2y + 1.

step 3: Solve for ?:

• x= 2y + 1
• Subtract 1 from both sides: x − 1 = 2y
• Divide both sides by 2: (x - 1)/2 = y

step 4: Replace y with ?^-1(x) : ?^-1(x) :(x - 1)/2

Question 2. Finding the Inverse of: f(x)=x²

Solution:

Replace f(x) with y: Equation: y = x²

Swap x and y: x = y ²

Solve for y: x^1/2 = y (Taking the square root, considering only the positive root for a one-to-one function)

Inverse Function: f ⁻¹(x)= x^1/2

Question 3. Finding the Inverse of: f(x) = 1/x

Solution:

Replace f(x) with y: Equation: y = 1/x

Swap x and y: x = 1/y

Solve for y: y = 1/x

Inverse Function: f −1(x) = 1/x

Question 4. Finding the Inverse of: f(x) = e^x

Solution:

Replace f(x) with y: Equation: y = e^x

Swap x and y: x = e^y

Solve for y: y=ln(x)

Inverse Function: f ⁻¹(x)= ln(x)

Question 5. Finding the Inverse of: f(x) = log(x) (Assuming base 10 logarithm)

Solution:

Replace f(x) with y: Equation: y = log(x)

Swap x and y: x = log(y)

Solve for y: y =10^x

Inverse Function: f ⁻¹(x) = 10^x

Read More,

• Relation and Function
  
• Domain and Range of Trigonometric Functions
• Range of a Function
• Hyperbolic Function

Conclusion

Understanding inverse functions is crucial for solving equations and analyzing relationships between quantities in mathematics. It allows for "undoing" operations performed by functions and gaining insights into their behavior. Inverse functions provide a deeper understanding of functions by reversing the input-output relationship. This two-way analysis is vital in fields like physics to understand cause and effect.

Additionally, inverse functions pave the way for composing functions, where the output of one function becomes the input for another, allowing for the solution of complex problems by breaking them down into smaller steps.