Function Representations

5 min read

Reference: Stewart §1.1

Why Functions Matter

Every time you check the weather app, you’re looking at a function: input a time, get a temperature. When your phone shows battery percentage dropping, that’s a function too. Functions are the mathematical language for describing how one quantity depends on another.

Understanding functions from multiple angles (words, numbers, pictures, and formulas) gives you flexibility in solving problems. Sometimes a graph reveals patterns that a formula hides. Sometimes an equation makes predictions that a table can’t.

Prerequisite Map

Quick Reference

Property	Value
Chapter	1 - Functions and Limits
Section	1.1
Difficulty	Beginner
Time	~20 minutes

Key Concepts

What Is a Function?

A function is a rule that assigns to each input exactly one output.

Think of a function as a machine: you feed in a number, the machine does something, and exactly one number comes out. If you could get two different outputs for the same input, the machine would be unpredictable, and that’s not a function.

Notation: We write $f(x) = $ (formula) where:

$f$ is the function’s name
$x$ is the input (also called the independent variable)
$f(x)$ is the output (also called the dependent variable)

The Four Representations

Representation	Description	Example
Verbal	Describe in words	“Square the input and add 3”
Numerical	Table of input-output pairs	See table below
Graphical	Plot points in the $xy$-plane	Parabola opening up
Algebraic	Formula	$f(x) = x^2 + 3$

Example: All four representations of the same function

Verbal: “Square the input and add 3.”

Numerical:

$x$	$f(x)$
$-2$	$7$
$-1$	$4$
$0$	$3$
$1$	$4$
$2$	$7$

Graphical: A parabola with vertex at $(0, 3)$, opening upward.

Algebraic: $f(x) = x^2 + 3$

The Vertical Line Test

How do you tell if a graph represents a function?

Draw (or imagine) vertical lines across the graph. If any vertical line crosses the graph more than once, the graph does not represent a function.

    Function (passes)           Not a function (fails)

         |                           |
       • |                         • | •
      /  |                        /  |  \
     /   |                       (   |   )
    -----+-----                  -----+-----
         |                           |
         |                           |

Why it works: A vertical line at $x = a$ hits all points with that $x$-value. If it hits twice, then $x = a$ produces two outputs, violating the function definition.

Common misconception

a function is only a formula.

This is the action-view-of-function error. A function is a correspondence between inputs and outputs -- the same input always produces the same single output. A formula is one way to describe that correspondence, but a table, a graph, or a verbal rule is equally valid. Saying “that table is not a function, there is no equation” misses the point. Table B above, where every input gives the same output 4, is a perfectly legitimate function (a constant function). The formula $f(x) = 4$ expresses the same rule, but the table stands on its own.

Common misconception

two different outputs for the same input means it is “almost” a function.

This is a form of the iconic-graph error, applied to tables. The sideways parabola $x = y^2 - 4$ looks like a parabola, and parabolas “look like functions,” so students sometimes say it is “basically” a function. But the vertical line test is absolute: any single input with two outputs disqualifies the relation entirely. There is no such thing as “almost a function” -- the relationship either assigns exactly one output to every input, or it does not.

Common misconception

the input column could be either column in a table.

This is the input-output-confusion error. By convention, the left column (or the top row in a horizontal table) lists the inputs, and the right column (or the bottom row) lists the outputs. When checking whether a table represents a function, scan the INPUT column for repeated values paired with different outputs. Scanning the output column for repeats checks the wrong thing: repeated outputs are allowed (two inputs can give the same output), but a repeated input with two different outputs is not allowed.

Practice Problems

Level 1 Recognizing Functions

Which of the following tables represents a function?

Table A:

$x$	$y$
1	5
2	7
3	9
2	11

Table B:

$x$	$y$
1	4
2	4
3	4
4	4

Thought Process

To check if a table represents a function, ask: “Does any input appear more than once with different outputs?”

Scan the $x$-column for repeated values. If the same $x$ gives different $y$ values, it’s not a function.

Show Answer

Table A: Not a function. The input $x = 2$ appears twice with different outputs ($y = 7$ and $y = 11$).

Table B: Is a function. Each input gives exactly one output. The fact that all outputs are the same (a constant function $y = 4$) doesn’t violate the definition.

Level 2 Converting Representations

A function is described verbally as: “Triple the input and subtract 7.”

Write the algebraic formula.
Create a table with inputs $x = -1, 0, 1, 2$.
Find $f(5)$.

Thought Process

Translate the words directly:

“Triple the input” means $3x$
“Subtract 7” means $- 7$

Then substitute each input value to build the table.

Show Answer

(a) $f(x) = 3x - 7$

(b)

$x$	$f(x) = 3x - 7$
$-1$	$3(-1) - 7 = -10$
$0$	$3(0) - 7 = -7$
$1$	$3(1) - 7 = -4$
$2$	$3(2) - 7 = -1$

(c) $f(5) = 3(5) - 7 = 15 - 7 = 8$

Level 3 Applying the Vertical Line Test

Consider the equation $x = y^2 - 4$.

Solve for $y$ to find all $y$-values when $x = 0$.
Does this equation define $y$ as a function of $x$? Explain.
What would the graph look like, and how does that relate to part (b)?

Thought Process

When checking if an equation defines a function, try to solve for the dependent variable. If one input can produce multiple outputs, it’s not a function.

The equation $x = y^2 - 4$ is a sideways parabola, and thinking about its shape helps visualize the vertical line test.

Show Answer

(a) Setting $x = 0$: $$0 = y^2 - 4$$ $$y^2 = 4$$ $$y = \pm 2$$

So when $x = 0$, we get $y = 2$ or $y = -2$.

(b) No, this does not define $y$ as a function of $x$. The single input $x = 0$ produces two different outputs ($y = 2$ and $y = -2$).

(c) The graph is a parabola opening to the right with vertex at $(-4, 0)$. A vertical line at $x = 0$ (or any $x > -4$) crosses the parabola twice, failing the vertical line test.

Level 4 Interpreting Real-World Functions

A ride-share app charges according to this rule: $4 base fare plus $2.50 per mile, rounded up to the nearest mile.

Write a verbal description and an algebraic formula for the cost $C$ as a function of miles $m$ (assume exact mileage for the formula).
A customer sees charges of \$9.00, \$14.00, and \$21.50 on three trips. What distances (in miles) could explain these costs?
Is it possible for two different distances to give the same cost using the exact formula? What about with rounding?

Thought Process

For part (a), identify the fixed cost and the rate.

For part (b), work backwards: $C = 4 + 2.50m$ means $m = (C - 4)/2.50$.

For part (c), think about what the rounding does to the function property.

Show Answer

(a)

Verbal: “Start with $4, then add $2.50 for each mile traveled.”
Algebraic: $C(m) = 4 + 2.50m$ (for exact mileage)

(b) Solving $C = 4 + 2.50m$ for $m$: $$m = \frac{C - 4}{2.50}$$

$C = 9.00$: $m = (9 - 4)/2.50 = 5/2.50 = 2$ miles
$C = 14.00$: $m = (14 - 4)/2.50 = 10/2.50 = 4$ miles
$C = 21.50$: $m = (21.50 - 4)/2.50 = 17.50/2.50 = 7$ miles

(c)

Exact formula: No, each distance gives a unique cost (linear functions are one-to-one).
With rounding: Yes! A 2.1-mile trip and a 2.8-mile trip both round to 3 miles, so both cost $4 + 2.50(3) = $11.50.

Level 5 When Can We Invert?

Consider a function $f$ given by a table:

$x$	$f(x)$
1	3
2	5
3	5
4	9

Is $f$ a function? Explain.
If we swap the columns (so inputs become outputs and vice versa), is the result a function? Explain.
What property must a function have for its "swapped" version to also be a function?
Give an example of a formula $g(x)$ where the swap works, and explain why.

Thought Process

Part (a): Check if each input has exactly one output.

Part (b): When you swap, the old outputs become new inputs. Check if any value appears twice in what’s now the input column.

Part (c): This leads to the concept of one-to-one functions (also called injective functions).

Part (d): Think of functions that never repeat output values: strictly increasing or decreasing functions work.

Show Answer

(a) Yes, $f$ is a function. Each input (1, 2, 3, 4) appears exactly once and produces exactly one output.

(b) The swapped table would be:

input	output
3	1
5	2
5	3
9	4

This is not a function because input 5 gives two different outputs (2 and 3).

(c) For the swap to be a function, the original function must be one-to-one (or injective): different inputs must always give different outputs. In other words, no output value can repeat.

(d) Example: $g(x) = 2x + 1$

This works because $g$ is strictly increasing. If $x_1 \neq x_2$, then $2x_1 + 1 \neq 2x_2 + 1$, so different inputs always give different outputs. The swapped function (the inverse) is $g^{-1}(x) = \frac{x-1}{2}$.

Mastery Checklist

I can state the definition of a function in my own words
I can convert between verbal, numerical, graphical, and algebraic representations
I can apply the Vertical Line Test to determine if a graph is a function
I can evaluate $f(a)$ for specific values using any representation
I understand that a function can have repeated outputs but not repeated inputs

Mental Model

The Vending Machine Analogy:

Think of a function as a vending machine:

Each button (input) gives you exactly one item (output)
Different buttons might give the same item (like $f(2) = f(3) = 5$), and that’s fine
But one button can’t give two different items, since that would break the machine (and the function definition)

Previous	Up	Next
Ch1 §1 Skills	Domain and Range

Last updated: 2026-01-22