Identity Function Definition Formula Graph and Solved Examples
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 Mathematics Explained Clearly
1. What is an identity function?
An identity function is a function that returns the same value as its input, meaning f(x) = x for all x in its domain. In other words, every element maps to itself.
- General form: f(x) = x
- Domain: Any set of numbers (real, integers, etc.)
- Range: Same as the domain
- Graph: A straight line passing through the origin
2. What is the formula for the identity function?
The formula for the identity function is f(x) = x. This means the output is exactly equal to the input.
- If x = 3, then f(3) = 3
- If x = −5, then f(−5) = −5
3. What does the graph of an identity function look like?
The graph of the identity function is a straight line with equation y = x. It has the following properties:
- Slope = 1
- Y-intercept = 0
- Passes through points like (0,0), (1,1), (−2,−2)
4. Is the identity function linear?
Yes, the identity function is a linear function because it can be written in the form f(x) = mx + b with m = 1 and b = 0. Specifically:
- Slope (m) = 1
- Y-intercept (b) = 0
5. Is the identity function one-to-one and onto?
Yes, the identity function is both one-to-one (injective) and onto (surjective) when the domain and codomain are the same set.
- Injective: If f(a) = f(b), then a = b
- Surjective: Every element in the codomain has a preimage
6. What is the inverse of the identity function?
The inverse of the identity function is the identity function itself, given by f⁻¹(x) = x. Since the original rule is f(x) = x:
- Swap x and y → x = y
- Solve for y → y = x
7. How do you prove a function is an identity function?
To prove a function is an identity function, you must show that f(x) = x for every x in its domain. Follow these steps:
- Start with the given function
- Simplify the expression algebraically
- Check if the result equals x
8. What is the domain and range of the identity function?
The domain and range of the identity function are the same set of values. If defined on real numbers, then:
- Domain = ℝ
- Range = ℝ
9. What is the difference between an identity function and a constant function?
The identity function outputs the same value as the input, while a constant function outputs the same fixed number for all inputs.
- Identity function: f(x) = x
- Constant function: f(x) = c, where c is a constant
10. Where is the identity function used in mathematics?
The identity function is used in algebra, calculus, and linear algebra as a function that leaves elements unchanged. Common uses include:
- As the identity element under function composition
- Defining identity matrices in linear algebra
- Understanding transformations that do not alter values
