Navigation: Wiki Home > Skills > Function Arithmetic

Function Arithmetic

Building Functions from Building Blocks

You already know how to add numbers. But what does it mean to add two functions? If $f(x) = x^2$ and $g(x) = \sqrt{x}$, what is $(f + g)(x)$?

Function arithmetic lets you combine functions using the same operations you use with numbers: addition, subtraction, multiplication, and division. The key insight is that you're creating a new function whose outputs are computed from the outputs of the original functions.

This skill matters because real-world quantities often depend on multiple factors combined together. Revenue minus cost gives profit. Velocity times time gives distance. Learning to combine functions prepares you for building mathematical models of complex systems.

Prerequisite Map

Quick Reference

Property	Value
Course	MATH 161
Chapter	1.3
Difficulty	Beginner
Time	~15 minutes

Key Concepts

The Four Operations

Given two functions $f$ and $g$, we define new functions as follows:

Operation	Definition	Domain
Sum	$(f + g)(x) = f(x) + g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Difference	$(f - g)(x) = f(x) - g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Product	$(fg)(x) = f(x) \cdot g(x)$	$\text{dom}(f) \cap \text{dom}(g)$
Quotient	$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$	$\text{dom}(f) \cap \text{dom}(g)$, where $g(x) \neq 0$

The Domain Rule

The domain of a combined function is where BOTH original functions are defined—and for division, the denominator can't be zero.

This is the most important concept in function arithmetic. Think of it as a Venn diagram:

Domain of f:         Domain of g:
   [────────]           [────────]
       ↓                    ↓
       └─────── ∩ ─────────┘
                ↓
         Domain of f ± g, fg
                ↓
         (minus where g = 0 for f/g)

Why Intersection?

To compute $(f + g)(x)$, you need both $f(x)$ and $g(x)$. If either function is undefined at some $x$, the sum is undefined there too.

Example: If $f(x) = \sqrt{x}$ (domain: $x \geq 0$) and $g(x) = \sqrt{4-x}$ (domain: $x \leq 4$), then:

$$\text{dom}(f + g) = [0, \infty) \cap (-\infty, 4] = [0, 4]$$

Common Domain Restrictions

Expression	Restriction	Reason
$\sqrt{\text{stuff}}$	$\text{stuff} \geq 0$	Can't take square root of negative
$\frac{1}{\text{stuff}}$	$\text{stuff} \neq 0$	Can't divide by zero
$\ln(\text{stuff})$	$\text{stuff} > 0$	Logarithm requires positive input

Working with Combined Functions

When simplifying combined functions:

1. Write out the definition
2. Substitute the formulas for $f(x)$ and $g(x)$
3. Simplify algebraically
4. State the domain (don't forget restrictions!)

Practice Problems

Mastery Checklist

• [ ] I can compute $(f \pm g)(a)$ for specific values
• [ ] I can find formulas for sums, differences, products, and quotients of functions
• [ ] I can determine the domain of combined functions using intersection
• [ ] I correctly exclude values where the denominator is zero
• [ ] I can handle functions with multiple restrictions (square roots AND fractions)
• [ ] I understand why the domain is the intersection of individual domains

Mental Model

The Assembly Line Analogy: Think of $f$ and $g$ as two workers on an assembly line. Each takes an input $x$ and produces an output. To combine their work:

• Sum: Both workers produce their output, then you add them together
• Product: Both produce output, then you multiply
• Quotient: Both produce output, then you divide (but if the bottom worker produces 0, the line breaks!)

The domain is where BOTH workers can do their job. If either worker can't handle an input, the combined operation fails.

---

Previous	Up	Next
Function Transformations	Section 1.3	Function Composition

---

Last updated: 2026-01-22