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Elementary Function — Definition, Formula & Examples

An elementary function is any function constructed from basic building blocks — polynomials, exponentials, logarithms, trigonometric functions, and their inverses — combined using addition, subtraction, multiplication, division, and composition.

A function ff is elementary if it can be expressed as a finite combination of constant functions, the identity function xx, the exponential function exe^x, and the logarithmic function ln⁡x\ln x, closed under the operations of addition, subtraction, multiplication, division, exponentiation, and composition, including all trigonometric and inverse trigonometric functions (since these can be expressed in terms of exponentials and logarithms over the complex numbers).

How It Works

To determine whether a function is elementary, check if it can be written entirely in terms of familiar function types: polynomials like 3x2+13x^2 + 1, rational functions like xx+1\frac{x}{x+1}, exponentials like 2x2^x, logarithms like ln⁡(x)\ln(x), and trig functions like sin⁡(x)\sin(x). Any finite chain of additions, multiplications, divisions, or compositions of these is still elementary. Functions that cannot be expressed this way — such as the error function erf(x)\text{erf}(x) or the gamma function Γ(x)\Gamma(x) — are called non-elementary or special functions.

Example

Problem: Determine whether f(x)=esin⁡(x)+ln⁡(x2+1)f(x) = e^{\sin(x)} + \ln(x^2 + 1) is an elementary function.
Identify components: The function contains sin⁡(x)\sin(x) (trigonometric), e(⋅)e^{(\cdot)} (exponential), ln⁡(⋅)\ln(\cdot) (logarithmic), x2+1x^2 + 1 (polynomial), and addition.
f(x)=esin⁡(x)+ln⁡(x2+1)f(x) = e^{\sin(x)} + \ln(x^2 + 1)
Check operations: The exponential is composed with sine, and the logarithm is composed with a polynomial. These parts are then added together. Composition and addition of elementary functions yield elementary functions.
Answer: Yes, f(x)=esin⁡(x)+ln⁡(x2+1)f(x) = e^{\sin(x)} + \ln(x^2 + 1) is an elementary function because it is built entirely from polynomials, exponentials, logarithms, and trigonometric functions combined through composition and addition.

Why It Matters

In calculus, knowing whether a function is elementary matters because some elementary functions have antiderivatives that are not elementary — for example, ∫e−x2 dx\int e^{-x^2}\,dx has no closed-form elementary answer. Recognizing elementary functions also helps you identify which differentiation and integration rules apply in AP Calculus and college math courses.

Common Mistakes

Mistake: Assuming every elementary function has an elementary antiderivative.
Correction: While every elementary function can be differentiated to produce another elementary function, the reverse is not true. The integral ∫e−x2 dx\int e^{-x^2}\,dx is a classic example of an elementary integrand whose antiderivative is non-elementary.