Inverse Functions

An inverse function is a function that reverses the effect of another function. If a function takes an input and gives an output, the inverse function takes that output and brings you back to the input.

Inverse functions are useful in solving equations, especially in algebra, trigonometry, and real-life problems like calculating speed, time, and distance in reverse.

 

Table of Contents

• Definition of Inverse Functions
• Graph of Inverse Functions
• How to Find the Inverse of a Function
• Types of Inverse Functions
  • Inverse Trigonometric Functions
  • Inverse Rational Function
  • Inverse Hyperbolic Functions
• Inverse Function Formula
• Finding Inverse Function Using Algebra
• Solved Examples of Inverse Functions
• Conclusion
• FAQs On Inverse Functions

 

Definition of Inverse Functions  

Inverse functions are mathematical functions that reverse the effect of a given function. If a function f maps an input x to an output y, then its inverse function, denoted by f⁻¹, maps y back to x.  

In simple terms, what is inverse function? It is a function that undoes the action of the original function. If f(x) = y, then f⁻¹(y) = x. This means:

• f(f⁻¹(x)) = x

• f⁻¹(f(x)) = x

For inverse functions to exist, the original function must be both one-to-one and onto.  

 

Graph of Inverse Functions  

The graph of inverse functions is a mirror image of the original function’s graph, reflected across the line y = x.  

Important points about the graph of inverse functions:  

• If (a, b) is a point on the graph of a function f(x), then (b, a) will be on the graph of its inverse function f⁻¹(x).  

• The line y = x is the axis of symmetry for a function and its inverse.  

• If a function passes the horizontal line test, it has an inverse.  
  

 

How to Find the Inverse of a Function  

To understand how to find the inverse of a function, follow these four algebraic steps:

1. Replace f(x) with y.

2. Interchange the variables x and y.

3. Solve the equation for y.

4. Replace y with f⁻¹(x).

This process works for basic algebraic functions, inverse rational functions, and sometimes inverse trigonometric functions when limited to specific domains.  

 

Example:

Given f(x) = 2x + 5,
Step 1: y = 2x + 5
Step 2: Swap x and y: x = 2y + 5
Step 3: Solve for y: y = (x - 5)/2
Step 4: Therefore, f⁻¹(x) = (x - 5)/2

 

Types of Inverse Functions  

There are several types of inverse functions based on the category of the original function. These include:  

1. Inverse Trigonometric Functions  

Inverse trigonometric functions help find angles when the values of trigonometric ratios are known. These include the inverses of sine, cosine, tangent, and their reciprocals.  

Common inverse trigonometric functions:

• sin⁻¹(x) – inverse of sine function

• cos⁻¹(x) – inverse of cosine function

• tan⁻¹(x) – inverse of tangent function

Examples:

• sin⁻¹(1/2) = π/6

• tan⁻¹(1) = π/4

The domains of inverse trigonometric functions are limited to keep the functions one-to-one and ensure they have valid inverses.  

 

2. Inverse Rational Function  

An inverse rational function is the inverse of a function that contains a ratio of polynomials. These functions are more complex to invert algebraically.  

Example:
Let f(x) = (2x + 3)/(x - 4)

To find the inverse rational function,
Step 1: Let y = (2x + 3)/(x - 4)
Step 2: Swap x and y: x = (2y + 3)/(y - 4)
Step 3: Multiply both sides by (y - 4): x(y - 4) = 2y + 3
Step 4: Expand and solve for y.

This is how to derive an inverse rational function step-by-step.  

 

3. Inverse Hyperbolic Functions  

Inverse hyperbolic functions are inverses of hyperbolic functions like sinh, cosh, and tanh.  

Common inverse hyperbolic functions:

• sinh⁻¹(x) – inverse hyperbolic sine

• cosh⁻¹(x) – inverse hyperbolic cosine

• tanh⁻¹(x) – inverse hyperbolic tangent
  
  

Examples:

• sinh⁻¹(x) = ln(x + √(x² + 1))

• tanh⁻¹(x) = 0.5 * ln((1 + x)/(1 - x))

These functions often appear in physics and engineering, especially in situations involving wave equations and fluid mechanics.  

 

Inverse Function Formula  

The inverse function formula provides a way to verify or find the inverse.  

If f(x) = y, then the inverse function formula is:  

• f⁻¹(y) = x  

To find an inverse with the inverse function formula, follow these steps:  

1. Replace f(x) with y  

2. Swap x and y  

3. Solve for y to get f⁻¹(x)  

 

Example using the inverse function formula:

Given: f(x) = (x - 3)/4

Step 1: y = (x - 3)/4
Step 2: Swap x and y: x = (y - 3)/4
Step 3: Multiply by 4: 4x = y - 3
Step 4: y = 4x + 3

Therefore, f⁻¹(x) = 4x + 3

This confirms that the inverse function formula gives you the inverse when applied step-by-step.  

 

Finding Inverse Function Using Algebra  

Finding inverse function with algebra involves switching dependent and independent variables through manipulation of expressions.  

Steps:  

1. Start with the function y = f(x)  

2. Switch x and y  

3. Solve the resulting equation for y  

4. Replace y with f⁻¹(x)  

This method works for polynomial functions, rational functions, and logarithmic functions.  

 

Algebra Example:  

Given f(x) = (x + 1)/(x - 2)

Step 1: Replace f(x) with y: y = (x + 1)/(x - 2)
Step 2: Swap variables: x = (y + 1)/(y - 2)
Step 3: Multiply both sides: x(y - 2) = y + 1
Step 4: xy - 2x = y + 1
Step 5: Group like terms: xy - y = 2x + 1
Step 6: Factor y: y(x - 1) = 2x + 1
Step 7: Solve: y = (2x + 1)/(x - 1)

So, f⁻¹(x) = (2x + 1)/(x - 1)

 

Solved Examples of Inverse Functions

Here are more examples showing how to find the inverse of a function.

Example 1:
f(x) = x² (Not invertible on full domain)
Restrict domain: x ≥ 0
Then:
y = x²
Swap: x = y²
Solve: y = √x
So: f⁻¹(x) = √x, for x ≥ 0

 

Example 2:
f(x) = 3x - 2
Inverse: f⁻¹(x) = (x + 2)/3

 

Example 3 (Inverse Rational Function):
f(x) = (x + 2)/(2x + 3)
Follow algebraic steps to find f⁻¹(x)

 

Example 4 (Inverse Trigonometric Functions):
If f(x) = sin(x), then f⁻¹(x) = sin⁻¹(x), but only for -1 ≤ x ≤ 1

 

Conclusion  

Understanding inverse functions is essential for solving algebraic and real-world problems. They help us reverse the process of a function, enabling solutions in fields like geometry, calculus, physics, and data modeling. From knowing what is inverse function to mastering how to find the inverse of a function, and applying the inverse function formula, each aspect strengthens your mathematical skills.

Focus on practicing problems involving inverse trigonometric functions, inverse rational function, and inverse hyperbolic functions to get a full grasp of this important concept.

 

Related Links

Trigonometric Identities - Learn essential trigonometric identities, including reciprocal, Pythagorean, and co-function identities, with proofs and examples for better understanding.

Trigonometry Formulas - Explore important trigonometric Formulas  covering ratios, identities, and angle transformations, with examples to help you apply them effectively.

Sin Cos Tan Values - Learn and memorize the standard sine, cosine, and tangent values for key angles to boost your trigonometry skills.

 

Frequently Asked Questions on Inverse Functions

1. How do you explain inverse functions?

Ans: Inverse functions reverse the operation of the original function. If f(x) gives y, then the inverse function f⁻¹(y) gives x. It “undoes” what the original function does.

2. How to find the inverse of a function?

Ans: To find the inverse of a function:

• Replace f(x) with y

• Swap x and y

• Solve for y

• Replace y with f⁻¹(x)

Example:
f(x) = 2x + 3
y = 2x + 3 → x = 2y + 3 → y = (x - 3)/2
So, f⁻¹(x) = (x - 3)/2

 

3. What is the inverse of 25?

Ans: If referring to the multiplicative inverse, it is 1/25.
If referring to an inverse function, context of the function is needed.

4. What does ∀ mean in math?

Ans: ∀ is a symbol meaning “for all.” For example, ∀x ∈ R means “for all x in the set of real numbers.”

 

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