Identity Function — Definition, Graph, Properties, and Examples
Book A Free Math Class
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 — 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 (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.
- Evaluate the identity function f(x)=x at x=−12.
- Do the points (7,7) and (7,3) lie on the identity function?
- Compute h∘I for h(x)=2x+1, where I(x)=x.
- Find the range of the identity function on the domain −1,0,4.
- 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.