Algebra of Functions

Just as numbers can be added, subtracted, multiplied, and divided, functions can be combined using the same arithmetic operations to create new functions. This branch of study is called the algebra of functions, and it is a foundational tool in mathematics that lets us build complex relationships from simpler ones. For example, if one function models the revenue of a business and another models its costs, subtracting the two gives a profit function — a real and powerful application of function arithmetic. When we operate on two functions f and g, the result is itself a new function whose output at any input x is determined by applying the operation to f(x) and g(x). The domain of the resulting function must be considered carefully: it is the set of all x values that belong to the domains of both original functions simultaneously — in other words, the intersection of their domains. If division is involved, any x values that make the denominator zero must also be excluded. Mastering the algebra of functions prepares you for calculus, where sums, products, and quotients of functions appear constantly in differentiation rules (sum rule, product rule, quotient rule). It also underpins the idea of function composition, where the output of one function feeds directly into another. Together, these tools give mathematicians and scientists enormous flexibility in modeling the real world.

Key takeaways
  • Identify the Two Functions and Their Domains
  • Addition of Functions: (f + g)(x)
  • Subtraction of Functions: (f − g)(x)
  • Multiplication of Functions: (f · g)(x)
  • Finding a Specific Value of a Combined Function
  • Determining the Domain of the Result

Identify the Two Functions and Their Domains

Before combining functions, write down f(x) and g(x) clearly, and state each function's domain. The domain of f is the set of all x for which f(x) is defined, and similarly for g. For example, if f(x) = √x, its domain is x ≥ 0, because you cannot take the square root of a negative number in the reals.

Addition of Functions: (f + g)(x)

The sum function is defined by (f + g)(x) = f(x) + g(x). Simply evaluate each function at x and add the results. The domain of (f + g) is the intersection of the domains of f and g — that is, Dom(f) ∩ Dom(g). Example: if f(x) = x² and g(x) = 3x − 1, then (f + g)(x) = x² + 3x − 1, with domain all real numbers.

Subtraction of Functions: (f − g)(x)

The difference function is defined by (f − g)(x) = f(x) − g(x). Subtract g(x) from f(x) for each x in the shared domain. Be careful with signs when g(x) is itself a polynomial — always use parentheses to distribute the minus sign. Example: (f − g)(x) = x² − (3x − 1) = x² − 3x + 1.

Multiplication of Functions: (f · g)(x)

The product function is defined by (f · g)(x) = f(x) · g(x). Multiply the expressions together and simplify. The domain is again Dom(f) ∩ Dom(g). Example: (f · g)(x) = x² · (3x − 1) = 3x³ − x². Products of functions arise naturally in physics, such as when force and displacement functions are multiplied to give work as a function of position.

Finding a Specific Value of a Combined Function

To evaluate a combined function at a specific number, substitute that number into the final simplified expression, or evaluate each original function separately first and then apply the operation. Both approaches yield the same answer. For instance, (f + g)(2) can be computed as f(2) + g(2) = 4 + 5 = 9, or by substituting into x² + 3x − 1: 4 + 6 − 1 = 9. Always verify using both methods when learning.

Determining the Domain of the Result

The domain of any combined function (sum, difference, or product) is Dom(f) ∩ Dom(g). If f(x) = √x (domain: x ≥ 0) and g(x) = √(4 − x) (domain: x ≤ 4), then (f + g)(x) = √x + √(4 − x) has domain [0, 4], because both functions must be defined simultaneously. Always state the domain explicitly — it is part of the complete definition of the new function.

The Key Equation

(f±g)(x)=f(x)±g(x),(f⋅g)(x)=f(x)⋅g(x),Dom=Dom(f)∩Dom(g)(f \pm g)(x) = f(x) \pm g(x), \quad (f \cdot g)(x) = f(x) \cdot g(x), \quad \text{Dom} = \text{Dom}(f) \cap \text{Dom}(g)
  • f(x) — First function (varies): The output of the first function evaluated at input x.
  • g(x) — Second function (varies): The output of the second function evaluated at input x.
  • (f \pm g)(x) — Sum or difference function (varies): The new function formed by adding or subtracting f and g pointwise.
  • (f \cdot g)(x) — Product function (varies): The new function formed by multiplying f and g pointwise.
  • \text{Dom}(f) \cap \text{Dom}(g) — Intersection of domains (set of real numbers): The set of all x values for which both f(x) and g(x) are defined; this becomes the domain of the combined function.

Worked Example

Problem. Let f(x) = 2x² − 3 and g(x) = x + 5. Find (f + g)(x), (f − g)(x), and (f · g)(x). Then evaluate each at x = 2.

(f+g)(x)=2x2+x+2,(f−g)(x)=2x2−x−8,(f⋅g)(x)=2x3+10x2−3x−15(f+g)(x)=2x^2+x+2,\quad (f-g)(x)=2x^2-x-8,\quad (f\cdot g)(x)=2x^3+10x^2-3x-15

Answer. (f + g)(x) = 2x² + x + 2, value at x = 2 is 12; (f − g)(x) = 2x² − x − 8, value at x = 2 is −2; (f · g)(x) = 2x³ + 10x² − 3x − 15, value at x = 2 is 35.

Figure
Diagram
Algebra of Functions: f(x)=x²−1, g(x)=x+2, and their sum (f+g)(x)=x²+x+1xy−9−7−5−3−11357f(x)=x²−1g(x)=x+2(f+g)(x)=x²+x+100
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler was the first mathematician to treat functions as central objects of study and to introduce the modern notation f(x) in 1734. His systematic approach to combining and manipulating functions laid the groundwork for what we now call the algebra of functions. He showed that complex mathematical relationships could be expressed and manipulated symbolically through functional notation, transforming how mathematicians communicate and reason about change. Euler lost sight in his right eye in his late twenties and became almost completely blind later in life, yet he produced roughly half of his total mathematical output after losing his sight — demonstrating extraordinary mental power for keeping entire calculations in his head.

  • Composition of Functions — Composition is a fifth operation on functions (beyond the four arithmetic ones) where the output of one function becomes the input of another — a natural next step after learning the algebra of functions.
  • Domain and Range — The domain of a combined function is the intersection of the domains of the component functions, so a solid understanding of domain and range is essential for the algebra of functions.
  • Introduction to Functions — Before combining functions arithmetically, you must understand what a function is, how it maps inputs to outputs, and how to use function notation — all covered in this foundational concept.
  • Operations on Polynomials — When the functions being combined are polynomials, the algebra of functions reduces to polynomial addition, subtraction, and multiplication — making this a direct application of polynomial operations.