Functions: 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.

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Learning Guide: The Coordinate System

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

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?

By mathematical convention, coordinates are always written in alphabetical order as

(x, y)

. This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.

What makes a relation a function?

A relation is a function if and only if each input value is associated with exactly one output value. If a single input has multiple different outputs, it is not a function.

How do you find where a function crosses the y-axis?

To find where a function crosses the

y

-axis, calculate

f(0)

by replacing

x

with

0

in the function formula. The resulting point on the graph will be

(0, f(0))

What happens if we input a value outside the domain?

If you input a value outside the domain, the function is undefined for that value. For example, in

f(x) = \frac{1}{x}

, inputting

x = 0

results in division by zero, which has no defined mathematical value.

How can you identify the domain of a function from its graph?

To find the domain from a graph, look at the graph's horizontal extent along the

x

-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).

What is the difference between domain and range?

The domain is the set of all valid input values (usually

x

) that you can feed into a function, while the range is the set of all output values (usually

y

) that the function produces as a result.

How can you identify the range of a function from its graph?

To find the range from a graph, look at the graph's vertical extent along the

y

-axis. Find the lowest and highest points of the graph, taking note of whether these endpoints are included (solid) or excluded (open circles).

What does it mean for a function to be continuous?

Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.

How do you find points where a function is not continuous?

Look for inputs that make the function undefined (like division by zero). For example,

f(x) = \frac{x^2 - 16}{x - 4}

is discontinuous at

x = 4

because you cannot divide by zero, creating a hole in the graph.

How do I determine whether a function is increasing or decreasing from its formula?

For linear functions

f(x) = mx + b

, the function increases if the slope

m > 0

and decreases if

m < 0

. For quadratic functions

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

, check the sign of

a

: if

a > 0

, the parabola opens up, so it decreases for

x < h

and increases for

x > h

Why can't I use y-values to define intervals of increase or decrease?

Although we look at the vertical rise or fall (y-values) to determine *what* the graph does, the intervals must specify *where* it happens horizontally. By convention, intervals of increase/decrease partition the domain of the function, which is represented by the

x

-axis.