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. 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 |
function arithmetic is the same as function composition.
This is the composition-is-not-chaining error applied in the other direction. $(f+g)(x) = f(x) + g(x)$ evaluates both functions at the SAME input $x$ and adds the results. $(f \circ g)(x) = f(g(x))$ feeds the output of $g$ as input to $f$ -- a completely different operation. With $f(x) = x^2$ and $g(x) = x+1$: the sum gives $(f+g)(3) = 9 + 4 = 13$, while the composition gives $(f \circ g)(3) = f(4) = 16$. The circle symbol $\circ$ signals composition; no symbol between $f$ and $g$ (or a $+$, $-$, or product dot) signals arithmetic on the outputs.
the domain of $f/g$ is the same as the domain of $f \cdot g$.
This is the input-output-confusion error applied to domain rules. For the quotient $f/g$, you need $g(x) \neq 0$ as an extra restriction beyond the intersection of the two domains. With $f(x) = \sqrt{x}$ and $g(x) = \sqrt{x-2}$, the product $fg$ has domain $[2, \infty)$. The quotient $f/g$ also excludes $x = 2$ because $g(2) = 0$, giving domain $(2, \infty)$. The two domains look almost identical but differ at one point. Forgetting this exclusion means the quotient formula produces a division by zero at $x = 2$, which is undefined.
Working with Combined Functions
When simplifying combined functions:
- Write out the definition
- Substitute the formulas for $f(x)$ and $g(x)$
- Simplify algebraically
- State the domain (don’t forget restrictions!)
Practice Problems
If $f(x) = 2x + 1$ and $g(x) = x^2$, find $(f + g)(3)$.
Let $f(x) = x^2 - 1$ and $g(x) = 2x + 3$. Find formulas for:
- $(f - g)(x)$
- $(fg)(x)$
Let $f(x) = \sqrt{x + 2}$ and $g(x) = x - 1$. Find the domain of $\left(\frac{f}{g}\right)(x)$.
Let $f(x) = \frac{1}{x-3}$ and $g(x) = \sqrt{x-1}$. Find the domain of $(fg)(x)$ and write a formula for this function.
Let $f$ be any function with domain symmetric about the origin (i.e., if $x$ is in the domain, so is $-x$).
Define: $$E(x) = \frac{f(x) + f(-x)}{2} \quad \text{and} \quad O(x) = \frac{f(x) - f(-x)}{2}$$
- Prove that $E$ is an even function (i.e., $E(-x) = E(x)$).
- Prove that $O$ is an odd function (i.e., $O(-x) = -O(x)$).
- Verify that $f(x) = E(x) + O(x)$.
- Apply this to $f(x) = e^x$. What are $E(x)$ and $O(x)$?
Mastery Checklist
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