# Functions, domains, and representations

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/functions-domains-and-representations

## Functions and Domains

A **function** assigns exactly one output to every input in its **domain**. Walk the input along the axis and $f(x)=\frac{1}{x-2}$ answers every time, until $x = 2$, where it asks for $1/0$ and has nothing to hand back.

$$f : D \subseteq \mathbb{R} \;\longrightarrow\; \mathbb{R}$$

- The **domain** $D$ collects every $x$ for which $f(x)$ is defined. For $f(x)=\frac{1}{x-2}$, we need $x \neq 2$, so $D = \mathbb{R}\setminus\{2\}$.
- The **range** is the set of all values $f$ actually takes.
- Number sets nest: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.

**Common pitfall:** A formula without its domain is only half a function. $f(x) = 1/x$ and "$1/x$ for $x > 0$" are different mathematical objects with different properties. Always state the domain before computing.

## Representations and Symmetry

Functions take different mathematical shapes depending on how they are written. 

| Form | Example |
| --- | --- |
| Explicit | $y = 3x + 2$ |
| Implicit | $x^{2}+y^{2}=9$ |
| Parametric | $(x,y)=(\cos t,\;\sin t)$ |

**Symmetry check:** A function is **even** when $f(-x)=f(x)$, meaning it is symmetric about the $y$-axis. A function is **odd** when $f(-x)=-f(x)$, meaning it is symmetric about the origin.

A formula without its explicit domain is an incomplete model. When checking properties, always verify both the algebraic rule and the underlying domain constraints to avoid false conclusions about range and symmetry.

## Practice questions

2 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A relation passing the vertical line test is a function; one passing the horizontal test as well is invertible. Why does invertibility need the second test?

A. Because the horizontal test checks that the domain is unrestricted
B. Because only continuous functions can be inverted, and the test detects continuity
C. Because the two tests are equivalent and the second is a confirmation
D. Because the inverse swaps inputs and outputs

**Answer:** D. Because the inverse swaps inputs and outputs

**Why:** Reflecting in $y = x$ turns the horizontal test into the vertical one. This is why $x^{2}$ needs its domain restricted before a square-root inverse exists.

Page: https://tryals.app/practice/physics-i/functions-domains-and-representations/a-relation-passing-the-vertical-line-test-is-a-function-one-passing

### 2. The same function can be given by a formula, a graph, a table or a description in words. Why do physicists move between the four rather than settling on one?

A. Because graphs are approximate and formulas are exact, so both are needed for a check
B. Because the four representations describe genuinely different functions
C. Because measurement produces tables and theory produces formulas, and they cannot be compared
D. Because each makes a different property immediately visible

**Answer:** D. Because each makes a different property immediately visible

**Why:** A representation is a tool, not the function. Reading a maximum off a graph and proving it from a formula are the same fact reached by different routes, and each route is cheap for some questions and expensive for others.

Page: https://tryals.app/practice/physics-i/functions-domains-and-representations/the-same-function-can-be-given-by-a-formula-a-graph-a-table-or-a