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