Identity Function

1. Concept Overview

An identity function is the simplest type of function where every input remains unchanged after mapping.

In other words, the function gives back the same value that is given as input.

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2. Definition of Identity Function

A function [f] defined by
[f(x)=x]
for all [x] in its domain is called an identity function.

Here:

• Input = Output
• Mapping is self-mapping

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3. Examples of Identity Function

• [f(x)=x]
• [g(t)=t]
• [h(y)=y]

In each case, the function returns the same element.

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4. Domain and Range of Identity Function

Domain:
The domain of an identity function is the given set itself.

Range:
The range is also the same set.

Thus:
Domain = Range

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5. Graph of Identity Function

The graph of [f(x)=x] is:

• a straight line
• passing through the origin
• making an angle of 45° with the positive x-axis

 

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6. Nature of Identity Function

An identity function is:

• one–one
• onto
• bijective
• continuous everywhere
• strictly increasing

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7. Important Properties

• Each element maps to itself
• Inverse of identity function is the identity function itself
• Slope of graph is 1
• Derivative of identity function is 1

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8. Identity Function vs Constant Function

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7. Conceptual Questions with Solutions

1. Why is it called an identity function?

Because each element is mapped to itself, preserving its identity.

2. Is [f(x)=x] a function?

Yes. Every input has exactly one output, satisfying the definition of a function.

3. Why is identity function one–one?

Because different inputs give different outputs.

4. Why is identity function onto?

Because every element of the codomain is an image of itself.

5. Can identity function be many–one?

No. It always gives distinct outputs for distinct inputs.

6. What is the inverse of identity function?

The inverse of identity function is the identity function itself.

7. Is identity function continuous?

Yes. It is continuous everywhere.

8. Why is its graph a straight line?

Because the function is linear with constant slope.

9. What is the slope of identity function?

The slope is 1.

10. Is identity function increasing?

Yes. As input increases, output also increases.

11. Can domain of identity function be restricted?

Yes. It can be defined on any subset of real numbers.

12. Is identity function polynomial?

Yes. It is a polynomial of degree one.

13. Why is identity function bijective?

Because it is both one–one and onto.

14. What happens if domain ≠ codomain?

Then the function may not be onto.

15. Where is identity function used?

It is used as a reference function in function composition.

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8. FAQ / Common Misconceptions

1. Identity function is same as constant function.

False. Identity function depends on input, constant function does not.

2. Identity function is many–one.

False. It is one–one.

3. Identity function has no inverse.

False. Its inverse exists and is itself.

4. Graph of identity function is horizontal.

False. It is a slant line.

5. Identity function is discontinuous.

False. It is continuous everywhere.

6. Identity function is decreasing.

False. It is strictly increasing.

7. Identity function must have domain R.

False. Domain can be any set.

8. Identity function is not algebraic.

False. It is an algebraic function.

9. Identity function is not useful.

False. It plays a key role in mathematics.

10. Identity function changes values.

False. It preserves values.

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9. Practice Questions with Step-by-Step Solutions

Question 1.

Find the domain and range of [f(x)=x].

Step-by-Step Solution:

1. Function is defined for all real [x].
2. Output equals input.

Conclusion:
Domain = [R]
Range = [R]

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Question 2.

State whether [f(x)=x] is one–one or many–one.

Step-by-Step Solution:

1. Different inputs give different outputs.

Conclusion:
The function is one–one.

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Question 3.

Find the inverse of [f(x)=x].

Step-by-Step Solution:

1. Write [y=x].
2. Interchange [x] and [y].

Conclusion:
Inverse is [f⁻¹(x)=x].

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Question 4.

Find [f′(x)] for [f(x)=x].

Step-by-Step Solution:

1. Derivative of [x] is [1].

Conclusion:
[f′(x)=1]

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Question 5.

Draw the graph of [f(x)=x].

Step-by-Step Solution:

1. Plot points [(0,0)], [(1,1)], [(−1,−1)].
2. Join them with a straight line.

Conclusion:
Graph is a straight line through origin.