Functions

Master the basics of coordinate systems, reading and writing points, identifying quadrants, and exploring function graphs.

Learning Guide: The Coordinate System

A coordinate system is a two-dimensional grid used to describe the exact position of points on a flat plane. It is formed by two number lines that cross each other.

Practice Topics

1. Coordinate System

Learn the two axes of a grid, the origin, and how to write and locate points.

2. Introduction to Functions

Understand what a function is, explore real-world analogies, master function notation f(x), and learn to identify points and y-intercepts.

3. Domain

Understand what the domain of a function is, learn about real-world input constraints (like radius and tax brackets), and see how domains are represented on graphs.

4. Range

Understand what the range of a function is, learn about special ranges (like square root, quadratic, and periodic functions), and practice reading ranges from graphs.

5. Continuity of a Function

Understand the concept of continuity, explore continuous and discontinuous functions, and practice finding points of discontinuity.

6. Increasing & Decreasing Functions

Learn how to identify intervals of increase and decrease on a graph, define domains using inequality notation, and understand the formal definition of monotonicity.

Learning Guide

1. Coordinate System

The Axes & The Origin

The grid is built from two intersecting axes:

x-axis: The horizontal axis. Its values increase as you move right (positive direction) and decrease as you move left (negative direction).
y-axis: The vertical axis. Its values increase as you move upwards (positive direction) and decrease as you move downwards (negative direction).
Intersection: The two axes cross each other at a 90° angle (they are perpendicular). The point where they cross is called the origin, represented by coordinates $(0, 0)$ .

Figure 1: The coordinate plane showing the x-axis, y-axis, origin, the four quadrants, and a plotted point at (3, 2).

Writing points as Ordered Pairs

Any point on the coordinate plane is represented as an ordered pair written in the format:

(x, y)

The first number ( $x$ ) is the x-coordinate (abscissa). It tells you how many units to move horizontally from the origin (right if positive, left if negative).
The second number ( $y$ ) is the y-coordinate (ordinate). It tells you how many units to move vertically from the origin (up if positive, down if negative).

Coordinates of a Point

(x, y)

x: horizontal position
y: vertical position

The Four Quadrants (Regions)

The two crossing axes divide the coordinate system into four regions called quadrants, numbered counter-clockwise starting from the top-right:

First Quadrant (I): Top-right region. Both $x$ and $y$ are positive ( $x > 0, y > 0$ ). Example: $(2, 3)$ .
Second Quadrant (II): Top-left region. $x$ is negative and $y$ is positive ( $x < 0, y > 0$ ). Example: $(-4, 1)$ .
Third Quadrant (III): Bottom-left region. Both $x$ and $y$ are negative ( $x < 0, y < 0$ ). Example: $(-2, -3)$ .
Fourth Quadrant (IV): Bottom-right region. $x$ is positive and $y$ is negative ( $x > 0, y < 0$ ). Example: $(3, -5)$ .

Note: Points that lie directly on the x-axis or y-axis (like

(0, 4)

(-3, 0)

) or at the origin

(0, 0)

are not located in any of the four quadrants.

Midpoint of a Segment

The midpoint

M

of a segment connecting points

A(x_1, y_1)

and

B(x_2, y_2)

is the point that lies exactly halfway between them. The coordinates of the midpoint are given by the formula card below:

Midpoint coordinates of a segment

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

A: (x₁, y₁), B: (x₂, y₂)

Mathematical Derivation & Example:
To understand why this formula works, we can project the segment

AB

and its midpoint

M

onto the coordinate axes. This splits both the horizontal and vertical intervals into two equal parts:

1. Horizontal Projection ( $x$ -axis):
The horizontal interval spans from

x_1

x_2

. On our graph, this goes from

x_1 = -3

x_2 = 5

(a total distance of

8

units). The midpoint

M

projects to

x_M = 1

. Since the distance from

x_1

x_M

must equal the distance from

x_M

x_2

x_M - x_1 = x_2 - x_M

Using our graph coordinates:

1 - (-3) = 5 - 1 = 4\text{ units}

Solving algebraically for

x_M

2x_M = x_1 + x_2 \implies x_M = \frac{x_1 + x_2}{2}

Substituting the values yields:

x_M = \frac{-3 + 5}{2} = \frac{2}{2} = 1

2. Vertical Projection ( $y$ -axis):
The vertical interval spans from

y_1

y_2

. On our graph, this goes from

y_1 = -2

y_2 = 4

(a total distance of

6

units). The midpoint

M

projects to

y_M = 1

. Equalizing the distances:

y_M - y_1 = y_2 - y_M

Using our graph coordinates:

1 - (-2) = 4 - 1 = 3\text{ units}

Solving algebraically for

y_M

2y_M = y_1 + y_2 \implies y_M = \frac{y_1 + y_2}{2}

Substituting the values yields:

y_M = \frac{-2 + 4}{2} = \frac{2}{2} = 1

By combining these components, we find the exact coordinates of the midpoint:

M(1, 1)

The graph shows that projecting the segment onto the

x

and

y

axes divides the interval between coordinates into two equal parts:

x_M - x_1 = x_2 - x_M

and

y_M - y_1 = y_2 - y_M

Distance Between Two Points

The distance

d

between two points

A(x_1, y_1)

and

B(x_2, y_2)

is the length of the straight line segment connecting them. The distance formula is given by the formula card below:

Distance between two points

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

d: Distance, A(x₁, y₁), B(x₂, y₂)

Relation to the Pythagorean Theorem:
By projecting the segment

AB

onto the axes, we construct a right triangle

ACB

with a horizontal leg

AC

(length

|x_2 - x_1|

) and a vertical leg

CB

(length

|y_2 - y_1|

).
Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), the hypotenuse

d

satisfies:

d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \implies d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

For example, using the points

A(-3, -2)

and

B(5, 4)

on the graph:
- The horizontal distance is

x_2 - x_1 = 5 - (-3) = 8

.
- The vertical distance is

y_2 - y_1 = 4 - (-2) = 6

.
- Applying the theorem:

d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

The graph shows the right triangle

ACB

formed by projecting the segment

AB

onto the axes, demonstrating how the distance formula is derived directly from the Pythagorean theorem.

2. Introduction to Functions

Learning Guide: Introduction to Functions

A function is a mathematical rule or relationship that connects inputs to outputs. For every input, a function assigns exactly one output.

Real-World Analogies

Vending Machine: You input a code (like $A1$ ), and the machine outputs exactly one specific snack (like potato chips). If one code could output different items at random, it wouldn't work like a function!
Temperature Over Time: For any specific hour of the day (input), there is exactly one specific temperature (output).
Buying Apples: If apples cost $3$ dollars per kilogram, the total price (output) depends on the weight (input). We can write this rule as: $Price = 3 \cdot Weight$ .

Function Notation $f(x)$

We write functions using the notation:

f(x) = \text{expression}

$f$ is the name of the function.
$x$ is the input variable.
$f(x)$ represents the output value (the result of the rule applied to $x$ ).

For example, if $f(x) = 2x + 1$ :
If we input $3$ , we substitute $x$ with $3$ : $f(3) = 2(3) + 1 = 7$ . The input $3$ gives the output $7$ .
If we input $0$ , we substitute $x$ with $0$ : $f(0) = 2(0) + 1 = 1$ .

Equations That Are Not Functions

It is important to understand that there are valid mathematical equations that are not functions. One of the simplest examples is

x = 1

(or

x = c

for any constant

c

). While

x = 1

is a perfectly valid and meaningful mathematical equation (describing a vertical line), it is not a function because the input

x = 1

corresponds to infinitely many output values

y

. The validity of a mathematical equation should not be confused with whether it defines a function.

Figure 1b: The vertical line

x = 1

is a valid mathematical equation but not a function, because the input

x = 1

corresponds to multiple different output values (like

y = 2

and

y = -2

Functions and Graphs

When we draw a function on a coordinate system, the input (

x

) corresponds to the horizontal position, and the output (

f(x)

y

) corresponds to the vertical position. Each input-output pair gives us a point on the graph:

If $f(3) = 5$ , it corresponds to the point $(3, 5)$ on the graph.
If $f(0) = 2$ , it corresponds to the point $(0, 2)$ on the graph.

Distinguishing Functions & the $y$ -axis Crossing

We can often distinguish functions by looking at where they cross the vertical

y

-axis. The point where a function crosses the

y

-axis is called the y-intercept. It is always found by setting the input to

0

, which gives the point

(0, f(0))

.

For example, look at the two lines in the graph below:

The blue line representing $f(x) = x + 1$ crosses the $y$ -axis at $(0, 1)$ , because $f(0) = 1$ .
The red line representing $g(x) = -x + 3$ crosses the $y$ -axis at $(0, 3)$ , because $g(0) = 3$ .
Since $f(0) \neq g(0)$ , they cross the vertical axis at different heights, helping us tell them apart!

Figure 1: Two functions f(x) = x + 1 (blue) and g(x) = -x + 3 (red), crossing the y-axis at f(0) = 1 and g(0) = 3 respectively.

3. Domain

Learning Guide: Domain of a Function

The domain of a function is the set of all possible input values (usually

x

-values) for which the function is defined and produces a valid real number.

Common Domain Restrictions

Domain constraints can arise from physical reality, defined rules, or mathematical operations that are not defined over the real numbers:

Circle Area (Physical Constraints): The area of a circle as a function of its radius is $A(r) = \pi r^2$ . Since a physical radius cannot be negative, the domain is restricted to $r \ge 0$ .
Tax Brackets (Piecewise Constraints): In income tax systems, tax rates are defined using brackets (piecewise intervals). For example: a 10% tax rate for income up to $5,000$ ( $0 \le x \le 5000$ ), and a 15% tax rate for income above $5,000$ ( $x > 5000$ ). Each bracket represents a part of the domain.
Rational Functions (Division by Zero): For the function $f(x) = \frac{1}{x - 3}$ , division by zero is undefined. Since the denominator cannot be $0$ , we set $x - 3 \neq 0$ , which gives a domain of all real numbers except $x = 3$ .
Radical Functions (Square Roots): For the function $g(x) = \sqrt{x - 2}$ , we cannot take the square root of a negative number in real mathematics. The expression under the square root must be non-negative, so $x - 2 \ge 0$ , which gives a domain of $x \ge 2$ .

Domains on Graphs & Boundaries

A function's domain can be determined by looking at the horizontal extent of its graph. If the domain starts or ends at a specific boundary, the way we draw the endpoint depends on whether the boundary is included:

Open Circle (small empty circle): Used for strict inequalities ( $>$ or $<$ ) where the boundary point itself is not included. For example, if the domain is $x > 5$ , the point at $x = 5$ is drawn as a small open circle.
Solid Point (normal filled point): Used when the boundary point is included ( $\ge$ or $\le$ ).

Figure 2: Comparison of endpoint styles on a graph. An open circle excludes the endpoint, while a solid point includes it.

4. Range

Learning Guide: Range of a Function

The range of a function is the set of all possible output values (usually

y

-values) obtained by evaluating the function over its entire domain.

Range of Square Root Functions

For the principal square root function

f(x) = \sqrt{x}

, the output is defined as the non-negative square root. Since we cannot obtain a negative number from a real square root, the range is restricted to non-negative numbers:

y \ge 0

(or

[0, \infty)

in interval notation).

Figure 3a: The square root function

f(x) = \sqrt{x}

yields non-negative outputs (

y \ge 0

Range of the Function $f(x) = x^2$

A quadratic function like

f(x) = x^2

takes any input and multiplies it by itself. Because the product of any real number with itself is never negative (e.g.,

(-3)^2 = 9

and

0^2 = 0

), the domain is all real numbers (

\mathbb{R}

), but the outputs are always non-negative. Therefore, its range is

y \ge 0

(or

[0, \infty)

Figure 3b: The quadratic function

f(x) = x^2

yields non-negative outputs (

y \ge 0

Range of Periodic Functions

Periodic functions repeat their values in cycles. For these functions, the output values cycle between a specific minimum and maximum value, meaning their range is bounded and forms a closed interval:

[y_{\text{min}}, y_{\text{max}}]

(or

y_{\text{min}} \le y \le y_{\text{max}}

). For example, the sine function,

f(x) = \sin(x)

, cycles continuously, restricting its range to

[-1, 1]

(don't worry if you aren't familiar with

\sin(x)

yet—it is only used here as an example).

Figure 3c: The periodic wave oscillates between a minimum and maximum boundary.

Mastering SealMath

When answering range questions, you can enter inequalities directly using keyboard shortcuts. The editor automatically converts them to math symbols:

To Write	Type on Keyboard	Math Display
Greater than or equal to	`>=`	$\ge$
Less than or equal to	`<=`	$\le$
Greater than	`>`	$>$
Less than	`<`	$<$

5. Continuity of a Function

Learning Guide: Continuity of a Function

A function

f(x)

is said to be continuous if its graph is a single connected curve with no breaks, holes, or jumps. Intuitively, you can think of a continuous function as one whose graph can be drawn without lifting your pencil.

Continuity of a Function

A function

f(x)

is continuous at a point

x = a

if:
1.

f(a)

is defined.
2. The graph approaches the same

y

-value from both the left and the right.
3. The function value at

a

matches this

y

-value (no jumps or holes).

Continuous and Discontinuous Functions

1. Continuous Functions

A function is continuous if its graph has no breaks, holes, or jumps. Intuitively, you can think of its graph as one that can be drawn without lifting your pencil.

Polynomial

Polynomials: Functions like $f(x) = x^2 - 3$ or $f(x) = 2x + 1$ are continuous everywhere.

Absolute Value

Absolute Value: $f(x) = |x - 2|$ is continuous everywhere.

2. Discontinuous Functions

A function is discontinuous if its graph has breaks, jumps, holes, or shoots off to infinity.

2.1 Jump Discontinuity

The function jumps from one height to another at a specific point. The left-hand limit and right-hand limit exist but are not equal.

Jump Discontinuity

Income Tax Brackets: An everyday example is tax rates, where the tax rate jumps (e.g., from 10% to 20%) once income crosses a certain threshold.

2.2 Hole Discontinuity (Removable)

The function is defined and continuous everywhere except at a single point, where there is a missing point (hole).

Hole Discontinuity

Rational Holes: $f(x) = \frac{x^2-4}{x-2}$ is undefined and has a hole at $x = 2$ , though it looks like the line $y = x + 2$ everywhere else.

2.3 Infinite Discontinuity (Vertical Asymptote)

The function shoots up or down towards infinity as it approaches a certain value, creating a vertical split in the graph.

Infinite Discontinuity

Unbounded Split: In $f(x) = \frac{1}{x-3}$ , as $x$ gets closer to 3, division by a tiny number makes the function value shoot up to $+\infty$ or down to $-\infty$ (where the symbol $\infty$ means infinity — growing larger and larger without limit). In mathematics, this vertical boundary line that the graph gets infinitely close to, but never touches, is called a vertical asymptote.
Because of this split, the function is discontinuous at $x = 3$ .

6. Increasing & Decreasing Functions

Learning Guide: Increasing & Decreasing Functions

A function is described as increasing, decreasing, or constant depending on how its output values (

y

) behave as its input values (

x

) move from left to right (as

x

increases).

Increasing/Decreasing Functions: A function is called an increasing function (or decreasing function) if it rises (or falls) across its entire domain. For example, a linear function $f(x) = 3x + 1$ is an increasing function everywhere.

Increasing Function (

f(x_2) > f(x_1)

)

Decreasing Function (

f(x_2) < f(x_1)

)

Intervals of Increase/Decrease: Many functions rise in some regions and fall in others (e.g., a parabola). For these functions, we define intervals (segments of the domain) where the function is increasing or decreasing.

Intervals: Decreasing for

x < 0

, Increasing for

x > 0

Formal Definitions

Let

x_1

and

x_2

be any two inputs in an interval or domain of the function:

Increasing: The behavior is increasing if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ . In simple terms, as you walk along the graph from left to right, you are going up.
Decreasing: The behavior is decreasing if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ . In simple terms, as you walk along the graph from left to right, you are going down.
Constant: The behavior is constant if $f(x_1) = f(x_2)$ for every pair of inputs (a flat horizontal line).

Definition of an increasing or decreasing function (over the entire domain):

Increasing & Decreasing Functions

A function

f

is:
• Increasing if

f(x_1) < f(x_2)

for all

x_1 < x_2

in its entire domain (always rises as

x

increases).
• Decreasing if

f(x_1) > f(x_2)

for all

x_1 < x_2

in its entire domain (always falls as

x

increases).

Definition of increasing or decreasing intervals (on specific intervals):

Intervals of Increase & Decrease

A function increases or decreases on a specific interval

I

if the relationship holds for all inputs in

I

:
• Increasing on $I$ :

f(x_1) < f(x_2)

for all

x_1 < x_2

I

.
• Decreasing on $I$ :

f(x_1) > f(x_2)

for all

x_1 < x_2

I

The Crucial Misconception: Domain vs. Range

The Pitfall: When students look at a graph to identify intervals where it increases or decreases, they naturally focus on the vertical axis (the y-values) because they are looking at the graph rising or falling. This leads to writing incorrect ranges like

y > 2

.

The Rule: Intervals of increase and decrease must always be specified using the horizontal axis (x-values). The vertical behavior tells us what the function is doing (rising or falling), but the

x

-values tell us where this behavior happens. For example, if a function increases to the right of

x = 5

, the correct interval is

x > 5

, not

y > 2

How to Write Intervals

Just like we did when defining domains, we use inequality notation to specify the horizontal intervals of the graph:

To the right of a boundary $a$ : If the behavior occurs for all inputs greater than $a$ , we write:
$x > a$
For example, if a function increases to the right of $x = 5$ , its interval of increase is $x > 5$ .
To the left of a boundary $a$ : If the behavior occurs for all inputs less than $a$ , we write:
$x < a$
For example, if a function decreases to the left of $x = -2$ , its interval of decrease is $x < -2$ .
Between two boundaries $a$ and $b$ : If the function is rising or falling between two values $a$ and $b$ , we write:
$a < x < b$

Analyzing U-Shaped and V-Shaped Curves

When given equations like

f(x) = a(x - h)^2 + k

, the graph forms a curved U-shape called a parabola with a turning point (vertex) at

x = h

.

To find where the parabola increases or decreases without graphing, look at the coefficient

a

(the multiplier in front):

Smiling Parabola ( $a > 0$ ): If $a$ is positive, the shape opens upwards like a smiling face. The vertex is the lowest point. The graph goes down first, then up. So, it is decreasing for $x < h$ and increasing for $x > h$ .
Sad Parabola ( $a < 0$ ): If $a$ is negative, the shape opens downwards like a sad face. The vertex is the highest point. The graph goes up first, then down. So, it is increasing for $x < h$ and decreasing for $x > h$ .

Smiling Parabola (

a > 0

)

Sad Parabola (

a < 0

)

Similarly, functions containing absolute values like

f(x) = a|x - h| + k

behave in the exact same way. The absolute value symbols

|x - h|

measure the distance from

h

, which creates a straight V-shape instead of a curved U-shape. The multiplier

a

determines if this V-shape opens upwards (standard V-shape) or downwards (upside-down V-shape):

Opens Upwards ( $a > 0$ ): The graph is a standard V-shape pointing down to its lowest point (vertex) at $x = h$ . It decreases first, then increases (similar to a smiling parabola).
Opens Downwards ( $a < 0$ ): The graph is an upside-down V-shape pointing up to its highest point (vertex) at $x = h$ . It increases first, then decreases (similar to a sad parabola).

Opens Upwards (

a > 0

)

Opens Downwards (

a < 0

)

Learning Topics

Learning Guide: The Coordinate System

Practice Topics

1. Coordinate System

2. Introduction to Functions

3. Domain

4. Range

5. Continuity of a Function

6. Increasing & Decreasing Functions

Learning Guide

1. Coordinate System

The Axes & The Origin

Writing points as Ordered Pairs

Coordinates of a Point

The Four Quadrants (Regions)

Midpoint of a Segment

Midpoint coordinates of a segment

Distance Between Two Points

Distance between two points

2. Introduction to Functions

Learning Guide: Introduction to Functions

Real-World Analogies

Function Notation f(x)f(x)f(x)

Equations That Are Not Functions

Functions and Graphs

Distinguishing Functions & the yyy-axis Crossing

3. Domain

Learning Guide: Domain of a Function

Common Domain Restrictions

Domains on Graphs & Boundaries

4. Range

Learning Guide: Range of a Function

Range of Square Root Functions

Range of the Function f(x)=x2f(x) = x^2f(x)=x2

Range of Periodic Functions

Mastering SealMath

5. Continuity of a Function

Learning Guide: Continuity of a Function

Continuity of a Function

Continuous and Discontinuous Functions

1. Continuous Functions

2. Discontinuous Functions

6. Increasing & Decreasing Functions

Learning Guide: Increasing & Decreasing Functions

Formal Definitions

Increasing & Decreasing Functions

Intervals of Increase & Decrease

The Crucial Misconception: Domain vs. Range

How to Write Intervals

Analyzing U-Shaped and V-Shaped Curves

Function Notation $f(x)$

Distinguishing Functions & the $y$ -axis Crossing

Range of the Function $f(x) = x^2$