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# Identity Function — Definition, Graph, Properties, and Examples

TL;DR

An identity function returns whatever you put in, unchanged — its rule is f(x)=x. This article defines the identity function, draws its graph as the straight line y=x through the origin, lists its key properties (it is its own inverse, bijective, slope 1), and works through six examples — including how it differs from a constant function.

## What Is an Identity Function?

An **identity function** is a function that always returns its input unchanged. For every element x in its domain, the output equals the input:

f(x)=x

It is sometimes written I(x) or id(x). Whatever value enters, the same value leaves — there is no scaling, no shift, no transformation. Because the output mirrors the input exactly, the **domain and range are identical**.

| Symbol | Meaning |
| --- | --- |
| f(x) or I(x) | the identity function |
| x | the input value (and, here, the output too) |
| y=x | the equation of its graph |

## What Does the Graph of an Identity Function Look Like?

The graph of f(x)=x is a **straight line through the origin** that makes a 45° angle with both axes. Its slope is always exactly 1, because output rises one unit for every one unit the input rises.

This line, y=x, is also the mirror used when graphing any [inverse relation](/content/math/algebra/relations-and-functions/index.html) — reflecting a graph across it produces the inverse. The identity function and that mirror line are the same object, which is no coincidence: the identity is its own inverse.

## What Are the Properties of an Identity Function?

The identity function carries a tidy set of properties that make it the "neutral element" of function composition.

- **Domain equals range.** Every output is its own input, so both sets are the same.

- **Slope is 1.** The graph rises at a constant 45°.

- **It is bijective.** It is both [one-to-one](/content/math/algebra/one-to-one-function/index.html) (no two inputs share an output) and onto (every value in the range is hit).

- **It is its own inverse.** Applying it, then applying it again, returns the start: f(f(x))=x. So f^{-1}=f.

- **It is the identity for composition.** For any function g, composing with the identity changes nothing: g∘I=g and I∘g=g.

## Examples of Identity Function

Six examples, from a single evaluation to composition and a domain over a finite set.

### Example 1

**Evaluate the identity function f(x)=x at x=9.**

The identity function returns its input.

f(9)=9

**Final answer:** 9.

### Example 2

**A student claims f(x)=1 is the identity function "because it never changes anything." Is that right?**

_Wrong attempt._ The student reasons that "identity" means "stays the same," and f(x)=1 always gives 1.

**Why it breaks.** Test an input. f(2) should equal 2 for an identity function, but f(2)=1≠2. The output did not match the input — so the value did change.

_Correct._ f(x)=1 is a **constant function**, not an identity function. The identity returns the _input_ unchanged; a constant returns the same _output_ regardless of input.

**Final answer:** f(x)=1 is constant; the identity function is f(x)=x.

### Example 3

**Show that the identity function is its own inverse.**

Apply f twice.

f(f(x))=f(x)=x

**Final answer:** f^{-1}=f, so the identity function is self-inverse.

### Example 4

**Let g(x)=3x−4. Compute g∘I and I∘g, where I(x)=x.**

I∘g means apply g, then I: I(g(x))=g(x)=3x−4.

g∘I means apply I, then g: g(I(x))=g(x)=3x−4.

**Final answer:** both equal 3x−4, so composing with the identity leaves g unchanged.

### Example 5

**Find the range of the identity function on the domain −2,0,5,11.**

Each element maps to itself.

f(−2)=−2, f(0)=0, f(5)=5, f(11)=11

**Final answer:** range = {-2, 0, 5, 11}, identical to the domain.

### Example 6

**A point sits at (6,6) on a graph. A second point sits at (6,2). Which lies on the identity function, and why?**

On the identity function, output equals input, so a point (a,b) lies on it only when a=b.

(6,6): here a=b, so it lies on y=x.

(6,2): here 6≠2, so it does not.

**Final answer:** (6,6) lies on the identity function; (6,2) does not.

## Why the Identity Function Matters

> "Every operation needs an element that changes nothing."

The identity function exists for the same reason 0 and 1 do — every system of combination needs a neutral element, and composition is no exception.

- **The neutral element of composition.** Composing any function with the identity leaves it untouched.

- **The baseline for transformations.** In graphics and geometry, the identity transformation is the "no movement" starting point.

- **A reference line.** The line y=x is the mirror for inverse functions and the diagonal against which symmetry is judged.

## Tripping Points to Avoid

### Mistake 1: Confusing the identity function with a constant function

**Where it slips in:** Reading the word "identity" as "unchanging output."

**Don't do this:** Treat f(x)=c as an identity function.

**The correct way:** The identity returns the _input_; a constant returns a fixed _output_. Test one value — they disagree immediately.

### Mistake 2: Thinking the identity function has slope 0

**Where it slips in:** Sketching the graph from memory.

**Don't do this:** Draw a flat horizontal line.

**The correct way:** The identity function has slope 1, a diagonal through the origin.

### Mistake 3: Forgetting the domain restricts the identity too

**Where it slips in:** Identity functions defined on a limited set.

**Don't do this:** Assume an identity function always covers all real numbers.

**The correct way:** The identity returns inputs unchanged _only on its stated domain_.

## Practice Questions

Try these, then check the answers below.

1. Evaluate the identity function f(x)=x at x=−12.
2. Do the points (7,7) and (7,3) lie on the identity function?
3. Compute h∘I for h(x)=2x+1, where I(x)=x.
4. Find the range of the identity function on the domain −1,0,4.
5. What is the inverse of the identity function, and what is its slope?

### Answers

**Answer to Question 1:** f(−12)=−12.

**Answer to Question 2:** (7,7) lies on it; (7,3) does not.

**Answer to Question 3:** h∘I leaves h unchanged.

**Answer to Question 4:** Range = {−1, 0, 4}, identical to the domain.

**Answer to Question 5:** The identity function is its own inverse, and its slope is 1.

## Conclusion

- An **identity function** returns its input unchanged: f(x)=x.

- Its graph is the straight line y=x through the origin, with slope 1 and a 45° angle to both axes.

- Its domain and range are identical, and it is bijective and its own inverse.

- It is the neutral element of function composition — composing with it changes nothing.

- The common trap is confusing it with a constant function.