Functions and Tables

Article Contents

What is a Function Table?
Building a Table from a Formula
Step-by-Step Example
Reading Values from a Table
Finding a Formula from a Table
Tables with Real-Life Data
From Table to Graph
Common Mistakes
Interactive Exercises

1. What is a Function Table? {#what-is-a-table}

A function table (or table of values) lists selected inputs and their corresponding outputs:

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

Each row contains an input-output pair $(x, f (x))$ .

Tables are especially useful for:

Computing specific values before drawing a graph
Organising measured or experimental data
Spotting patterns in a function's behaviour

2. Building a Table from a Formula {#building-a-table}

Given a formula, follow these steps:

Choose several $x$ -values (include negative, zero, and positive values)
Substitute each $x$ into the formula
Calculate $f (x)$
Write the results in the table

Example: $f (x) = x^{2} + 3$

$x$	Calculation	$f (x)$
$- 3$	$(- 3)^{2} + 3 = 9 + 3$	$12$
$- 2$	$(- 2)^{2} + 3 = 4 + 3$	$7$
$- 1$	$(- 1)^{2} + 3 = 1 + 3$	$4$
$0$	$(0)^{2} + 3 = 0 + 3$	$3$
$1$	$(1)^{2} + 3 = 1 + 3$	$4$
$2$	$(2)^{2} + 3 = 4 + 3$	$7$
$3$	$(3)^{2} + 3 = 9 + 3$	$12$

3. Step-by-Step Example {#step-by-step}

Task: Complete the table for

f (x) = 2 x - 4

. Step 1: Choose values:

x \in {- 2, - 1, 0, 1, 2, 3, 4}

Step 2: Substitute and calculate:

f (- 2) = 2 (- 2) - 4 = - 4 - 4 = - 8

f (- 1) = 2 (- 1) - 4 = - 2 - 4 = - 6

f (0) = 2 (0) - 4 = 0 - 4 = - 4

f (1) = 2 (1) - 4 = 2 - 4 = - 2

f (2) = 2 (2) - 4 = 4 - 4 = 0

f (3) = 2 (3) - 4 = 6 - 4 = 2

f (4) = 2 (4) - 4 = 8 - 4 = 4

Step 3: Write the table:

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$
$f (x)$	$- 8$	$- 6$	$- 4$	$- 2$	$0$	$2$	$4$

Notice the pattern: each time $x$ increases by 1, $f (x)$ increases by 2. This is because the coefficient of $x$ is 2.

4. Reading Values from a Table {#reading-from-table}

Given a table, you can answer questions like:

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$f (x)$	$5$	$3$	$1$	$- 1$	$- 3$	$- 5$

Q: What is $f (3)$ ?

Look at the column where $x = 3$ . The answer is $f (3) = - 1$ .

Q: For which $x$ is $f (x) = 1$ ?

Find the column where $f (x) = 1$ . The answer is $x = 2$ .

Q: For which $x$ is $f (x) = 0$ ?

The value $f (x) = 0$ does not appear in the table. It falls between $x = 2$ and $x = 3$ . We would need the formula or graph to find the exact value.

5. Finding a Formula from a Table {#formula-from-table}

Sometimes you need to discover the rule behind a table.

Example

$x$	$0$	$1$	$2$	$3$	$4$
$f (x)$	$1$	$4$	$7$	$10$	$13$

Step 1: Find the pattern in outputs:

4 - 1 = 3

7 - 4 = 3

10 - 7 = 3

The difference is constant ( $+ 3$ ), so this is a linear function: $f (x) = k x + q$ .

Step 2: The constant difference gives us the slope:

k = 3

. Step 3: Use a known point, e.g.

f (0) = 1

1 = 3 (0) + q ⟹ q = 1

Result:

f (x) = 3 x + 1

Verification

f (4) = 3 (4) + 1 = 13

-- matches the table.

6. Tables with Real-Life Data {#real-life-tables}

Temperature throughout the day

Time	6:00	9:00	12:00	15:00	18:00	21:00
Temp (C)	$8$	$14$	$21$	$23$	$18$	$12$

Here the independent variable is time and the dependent variable is temperature.

Taxi fare

Distance (km)	$0$	$1$	$2$	$3$	$5$	$10$
Fare (\$)	$3$	$5$	$7$	$9$	$13$	$23$

The fare increases by \ $2 per km, with a \$ 3 base fee: $f (d) = 2 d + 3$ .

7. From Table to Graph {#table-to-graph}

A table provides the points you need to draw a graph:

Each pair $(x, f (x))$ becomes a point in the coordinate plane
Plot all points
Connect them with a smooth curve (or a straight line if linear)

See the detailed process in Functions and Graphs.

8. Common Mistakes {#common-mistakes}

Mistake	How to avoid it
Forgetting negative signs when substituting	Use parentheses: $f (- 2) = (- 2)^{2}$ , not $- 2^{2}$
Choosing only positive $x$ -values	Always include negative values and zero
Assuming a linear pattern without checking	Verify the difference is truly constant
Skipping the calculation column	Write out intermediate steps to reduce errors

9. Interactive Exercises {#interactive-exercises}

Practice working with function tables:

Summary

Concept	Description
Function table	Lists input-output pairs $(x, f (x))$
Building a table	Choose $x$ -values, substitute into the formula, compute $f (x)$
Reading a table	Find the column with the given $x$ or $f (x)$
Finding a formula	Look for constant differences (linear) or other patterns

Introduction to Functions -- what is a function
Variables in Functions -- independent and dependent variables
Functions and Graphs -- plotting tables as graphs
Linear Function -- the most common table pattern

Functions and Tables

Functions and Tables

Article Contents

1. What is a Function Table? {#what-is-a-table}

2. Building a Table from a Formula {#building-a-table}

Example: f(x)=x2+3

3. Step-by-Step Example {#step-by-step}

4. Reading Values from a Table {#reading-from-table}

5. Finding a Formula from a Table {#formula-from-table}

Example

Verification

6. Tables with Real-Life Data {#real-life-tables}

Temperature throughout the day

Taxi fare

7. From Table to Graph {#table-to-graph}

8. Common Mistakes {#common-mistakes}

9. Interactive Exercises {#interactive-exercises}

Summary

Related Articles

Example: $f (x) = x^{2} + 3$