What Is the Identity Function?

The identity function maps every input to itself. It is usually written as $I(x)=x$, so applying it does not change the value of $x$.

For example, if the domain is the real numbers, then

$$I(-4)=-4,\qquad I(2)=2,\qquad I(7.5)=7.5.$$

Graphically, the identity function is the straight line $y=x$. This graph is also important because the graphs of a function and its inverse are reflections of each other in $y=x$.

More precisely, an identity function is defined on a particular set. If $A$ is the set, write $I_A:A\to A$, where $I_A(x)=x$ for every . This matters because a composition is defined only when the output of the inner function lies in the domain of the outer function. Thus, uses the identity on the domain of , while uses the identity on its codomain .

Identity Under Composition

In IB Mathematics AA SL 2.5, the identity function is closely connected to composite functions. For a function $f$, composing with the appropriate identity function leaves $f$ unchanged:

$$(f\circ I)(x)=f(I(x))=f(x)$$

and

$$(I\circ f)(x)=I(f(x))=f(x).$$

It also provides the defining test for an inverse function. If $f^{-1}$ is the inverse of $f$, then

$$(f\circ f^{-1})(x)=x$$

and

$$(f^{-1}\circ f)(x)=x.$$

For example, let $f(x)=3x-5$. Its inverse is $f^{-1}(x)=\frac{x+5}{3}$. Therefore,

$$f\left(f^{-1}(x)\right)=3\left(\frac{x+5}{3}\right)-5=x.$$

Similarly, $f^{-1}(f(x))=\frac{(3x-5)+5}{3}=x$, so both required compositions give the identity on the relevant sets and confirm that each function reverses the other completely.

A common misconception is that $f^{-1}(x)=\frac{1}{f(x)}$. This is false: the superscript $$ denotes the inverse function, not the reciprocal.

IB Exam Technique

For an SL 2.5 Paper 1 or Paper 2 question asking you to verify an inverse, calculate both compositions and show that each simplifies to $x$, while respecting the relevant domains. Do not simply state that the functions are inverses without showing the composition.