Combining Functions

Definition

New functions can be created by performing arithmetic operations on existing functions $f$ and $g$. For all $x$ in the intersection of their domains ($D(f) \cap D(g)$):

• $(f+g)(x) = f(x) + g(x)$
  • How to read: “The function f plus g of x equals f of x plus g of x.”
  • Meaning: Add the output values point by point (superposition).
• $(f-g)(x) = f(x) - g(x)$
  • How to read: “The function f minus g of x equals f of x minus g of x.”
  • Meaning: Subtract outputs at each shared input (e.g., profit = revenue − cost).
• $(fg)(x) = f(x)g(x)$
  • How to read: “The function f times g of x equals f of x times g of x.”
  • Meaning: Multiply outputs at each $x$ (e.g., power = current × voltage).
• $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$.
  • How to read: “The function f divided by g of x equals the ratio of f of x to g of x, provided g of x is not zero.”
  • Meaning: Divide outputs pointwise; exclude $x$ where the denominator vanishes.

Why It Matters

It provides the modularity required to build complex models of reality from simple, well-understood mathematical building blocks.

Core Concepts

• Domain Intersection: The most critical constraint is that a combined function is only valid where both original functions are defined.
• Quotient Restriction: The quotient function inherits the domain intersection but must further exclude any $x$ that causes the denominator $g(x)$ to be zero.
• Pointwise Addition: Visually, $(f+g)$ is constructed by adding the $y$-coordinates of $f$ and $g$ at every $x$ in the shared domain.

Connected Concepts

• Function Definition, Function Notation
• Function Domain and Function Range
• Graphs of Functions: Properties