Identity Function

High School

Definition

A function that always gives you the same value as its argument. That means you're working with a function that looks like f(x) = x, where every real number x that is inputted gives you that same number x as the output. The graphs of identity functions all looks like a straight diagonal line that rises upwards from left to right.

Worked examples

\(f(x) = x\)

Whatever input you choose, the output is identical—this is the identity function.

\(f(5) = 5, \quad f(-3) = -3, \quad f(0) = 0\)

Each input returns itself as the output; the function changes nothing.

\(y = x\) graphs as a straight line through the origin with slope 1

The diagonal line shows every point \((a, a)\) lying on the graph.

Common mistakes

\(f(x) = x + 1\) is an identity function → \(f(x) = x\) is the identity function Adding or changing x in any way breaks the identity property—output must equal input exactly.
The identity function has slope zero → The identity function \(y = x\) has slope 1 The graph rises diagonally at 45°, not horizontally.
\(f(x) = |x|\) is the identity function → \(f(x) = x\) is the identity function Absolute value changes negative inputs to positive, so it does not return the input unchanged.

Where you'll use it next

The identity function is the neutral element in function composition, appears as the base case in transformations and inverse-function work, and serves as a building block when you study linear algebra and mappings.

Found in 1 StudyPug lesson

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