What are the Properties and Graph of the Identity Function?
The concept of identity function plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It helps students easily recognize patterns in functions and strengthens foundational knowledge for higher-level Maths.
What Is Identity Function?
An identity function is a special type of function where the output is always equal to the input. In mathematical terms, for every real number x, the identity function is defined as f(x) = x. You’ll find this concept applied in algebra, sets and relations, and functions, and it forms the basis for understanding mapping and transformations in higher maths.
Key Formula for Identity Function
Here’s the standard formula: \( f(x) = x \)
For all x in the set of real numbers (x ∈ ℝ).
Properties of Identity Function
| Property | Details |
|---|---|
| Domain | All real numbers (ℝ) |
| Range | All real numbers (ℝ) |
| Graph Shape | Straight line passing through the origin with slope 1 |
| Intercept | Origin (0,0) |
| Inverse | Self-inverse (identity is its own inverse) |
Graph of Identity Function
The graph of the identity function is a straight line making a 45° angle with both axes and passing through the origin (0,0). Every point (x, x) lies on this line. This helps you easily recognize and sketch the identity function during exams.
| x | f(x) |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Identity Function Example (Step-by-Step)
Let’s check if f(a) = a for inputs a = -3, 0, 2, and 5:
1. f(-3) = -32. f(0) = 0
3. f(2) = 2
4. f(5) = 5
In each case, output equals input, so this is the identity function.
Identity Function vs Other Functions
| Type | Formula | Output Pattern |
|---|---|---|
| Identity Function | f(x) = x | Output always equals input |
| Constant Function | f(x) = k | Output is the same constant, whatever the input |
| Linear Function | f(x) = mx + c | Slope m, intercept c; output changes with input |
Applications and Uses
The identity function is not only useful in Maths but also plays an important role in Physics, Computer Science, and logical reasoning. For example, mapping data values, designing identity matrices in algebra, and programming logic in Python or SQL often use the concept of identity. Students preparing for JEE or NEET will see its relevance in various questions about functions, transformations, and inverses.
Practice Problems: Try These Yourself
- State the domain and range of the identity function.
- Check if f(x) = 4x is an identity function for all real x.
- Write two real-world situations using the identity function.
- Sketch the graph of f(x) = x and f(x) = 0 on the same axes.
Frequent Errors and Misunderstandings
- Confusing the identity function with the constant function.
- Forgetting that the identity function’s range is also all real numbers (not just natural numbers).
- Drawing the incorrect graph (not a 45° line through the origin).
Relation to Other Concepts
The idea of identity function connects closely with topics such as linear functions and constant functions. Understanding identity also helps you grasp function inverses and algebraic mappings, making higher level Maths easier.
Classroom Tip
A quick way to remember identity functions: “Input equals output.” Think of a mirror—the number you put in, you see exactly the same number out! Vedantu’s teachers often use number mapping games to help make these ideas visual and fun in class.
We explored identity function—from definition, formula, properties, examples, mistakes, and how it links to other Maths topics. Continue practicing with Vedantu to become confident in solving function-based problems, and revisit function types and definitions for a stronger foundation.
Related Readings:
FAQs on Identity Function in Maths: Formula, Graph & Key Facts
1. What is the identity function in mathematics?
The identity function, often denoted as f(x) = x, is a fundamental function where the output value is identical to the input value for all elements in the domain. In simpler terms, it 'returns' the same value it receives as input. This makes it a crucial concept in various mathematical fields.
2. What is the equation or formula for the identity function?
The standard formula for the identity function is f(x) = x. This means that for any value of x, the function f(x) will always equal x itself. Other notations may exist, but this is the most common and universally understood representation.
3. What are the domain and range of the identity function?
The domain and range of the identity function are typically the set of all real numbers (ℝ). This means the function accepts any real number as input and produces the same real number as output. However, the domain and range can be restricted depending on the context or specific application.
4. How is the identity function graphed?
The graph of the identity function f(x) = x is a straight line passing through the origin (0,0) with a slope of 1. The line makes a 45-degree angle with both the x-axis and the y-axis because the y-coordinate always equals the x-coordinate.
5. What is the difference between an identity and a constant function?
A constant function always returns the same output value regardless of the input, e.g., f(x) = 5. The identity function, conversely, returns the input value itself. The key difference lies in the output: constant functions have a fixed output, while identity functions have a variable output that mirrors the input.
6. Why is it called an “identity” function?
It's called the 'identity' function because it leaves the input unchanged; it's the identity mapping of elements to themselves. It's like a mirror reflecting the input without any transformation or alteration.
7. How does the identity function behave when composed with other functions?
When the identity function is composed with another function (either before or after), it leaves the other function unchanged. For example, if g(x) is any function, then (f ∘ g)(x) = f(g(x)) = g(x) and (g ∘ f)(x) = g(f(x)) = g(x). It acts as a neutral element in function composition.
8. Is the identity function an inverse of itself?
Yes, the identity function is its own inverse. The composition of the identity function with itself results in the identity function again: (f ∘ f)(x) = f(f(x)) = x.
9. What are some real-world applications of identity functions?
While seemingly simple, identity functions have applications in various fields. They're fundamental in programming (e.g., in identity transformations), computer graphics (as a neutral transformation), and various areas of mathematics, serving as a building block for more complex functions and operations.
10. How is the identity function represented in discrete mathematics?
In discrete mathematics, the identity function can be represented as a mapping where each element in a set maps to itself. It's often used in set theory, graph theory, and other areas dealing with discrete structures. For example, in a set A = {1, 2, 3}, the identity function would be {(1,1), (2,2), (3,3)}.
11. Can the identity function be defined on non-real number sets?
Yes, the identity function's concept extends beyond real numbers. It can be defined on any set, whether it's a set of complex numbers, matrices, or other mathematical objects. The core principle remains the same: each element maps to itself.
12. What are some examples of identity functions in different contexts?
• In real numbers: f(x) = x
• In matrices: An identity matrix (a square matrix with 1s on the diagonal and 0s elsewhere) acts as an identity function under matrix multiplication.
• In set theory: The identity relation {(a,a) | a ∈ A} on a set A.
