Interval Notation Increasing And Decreasing

In this explainer, we will learn how to find the intervals over which a function is increasing, constant, or decreasing.

Throughout this explainer, we will use interval note to describe the intervals of increment and decrease. We begin past recalling what nosotros hateful by interval notation.

Definition: Interval Notation

The interval of numbers between and , including and , is denoted past , where and are called the endpoints of the interval.

To indicate that i of the endpoints is to be excluded from the set, the corresponding square bracket is inverse. Thus, in set up-builder annotation, this is expressed as follows:

We will now define the conditions for any function to be increasing, decreasing, or constant over a given interval.

Definition: Increasing Functions

A function is increasing on an interval if for whatever in .

Since nosotros need to compare to , the function must be defined on . When a part is increasing on an interval, its outputs are increasing on this interval, so its curve must be rising on this interval.

Definition: Decreasing Functions

A office is decreasing on an interval if for any in .

When a part is decreasing on an interval, its outputs are decreasing on this interval, and then its curve must be falling on this interval.

It is also common to refer to functions equally strictly increasing or strictly decreasing; however, we volition not be using this terminology in this explainer.

We can besides describe functions that do non modify outputs on an interval as follows.

Definition: Constant Function on an Interval

A function is constant on an interval if for any in : , for some constant .

When a function is constant on an interval, its outputs are constant on this interval, then its graph will be horizontal on this interval.

Definition: Increasing, Decreasing, or Abiding Functions

If a function is increasing on its entire domain, we only say the function is increasing. Likewise, if a function is decreasing on its entire domain, we but say the function is decreasing. Finally, if a role is constant on its unabridged domain, we just say the role is abiding.

We will now consider a multifariousness of graphs and determine on which intervals in their domain the functions are increasing, decreasing, or constant. In the start example, we volition decide this information from a given graph of the function.

Case 1: Identifying Whether the Part In the Given Graph is Increasing, Decreasing, or Neither

The graph of a function is given below. Which of the following statements about the function is true?

The office is abiding on .
The function is increasing on .
The function is decreasing on .
The function is constant on .
The function is increasing on .

Reply

Let u.s. begin by recalling what the words increasing, decreasing, and abiding tell us about the graphs of a office.

Firstly, a function is increasing on an interval if for any in . Secondly, a function is decreasing on an interval if for any in . Finally, a function is constant on an interval if for whatsoever in , for some constant .

If we compare these definitions to our graph, nosotros see that we have a horizontal line, so our part must be constant. For each value of , we meet that the output, , is equal to .

If nosotros await at the horizontal line representing our function, we see that information technology has arrows at both ends. This means that the line must extend from to . The interval is the ready of all real numbers; therefore, the function must be constant for all real numbers.

Therefore, our answer is selection A; the office is abiding on .

In our side by side case, we volition use the graph of a role to determine the intervals over which information technology is increasing, decreasing, or constant.

Example 2: Describing the Monotonicity of a Piecewise Function Using a Graph

Which of the post-obit statements correctly describe the monotonicity of the function represented in the figure below?

The part is increasing on and decreasing on .
The function is increasing on and decreasing on .
The office is increasing on , constant on , and decreasing on .
The function is increasing on , constant on , and decreasing on .

Answer

The monotonicity of a role describes whether it is increasing or decreasing over a given interval. In this case, we are given four possible options.

We think that a office is increasing on an interval if for any in . A function is decreasing on an interval if for any in . Finally, a function is abiding on an interval if for any in : , for some constant .

Our graph has three main sections.

Between and , every bit the value of increases, the output, , also increases. This means that the function is increasing on . As the options use open up intervals, nosotros practice not need to worry about the endpoints of the intervals in this question.

Between and , the output, , is ever equal to 3, so the role is constant on this interval. This ways that the function is constant on . Once again, our interval is open at both ends.

Between and , equally the value of increases, the output, , decreases. This ways that the function is decreasing on .

The monotonicity of the function can exist described as increasing on , abiding on , and decreasing on .

Therefore, our respond is option C; the function is increasing on , constant on , and decreasing on .

In our next example, we will place increasing and decreasing regions from a reciprocal graph.

Case 3: Identifying the Increasing and Decreasing Regions of a Graph

The graph of a part is given below. Which of the following statements well-nigh the role is true?

The function is increasing on and .
The function is decreasing on and .
The part is increasing on and .
The part is decreasing on and .

Reply

Each of the statements considers the monotonicity of the function; that is, whether it is increasing or decreasing over a given interval. We recall that a part is increasing on an interval if for any in . A function is decreasing on an interval if for any in .

Our graph has two asymptotes. We encounter that the -axis is a vertical asymptote and we have a horizontal asymptote at . This means that 0 is not in the domain of .

As 0 is not in the domain of , we need to consider the monotonicity of the part on its domain, the intervals and .

Let u.s.a. at present consider what happens to our graph as increases. As nosotros motion from to 0 forth the -axis, the value of increases. This means that on the interval , the office is increasing.

The aforementioned affair occurs as we move from 0 to along the -axis; the outputs of are increasing. This ways that on the interval , the part is too increasing.

It is important to note what is happening at . 0 is not in the domain of , which means that it is non in the intervals where is increasing or decreasing.

We can conclude that the function is increasing on and .

Equally 0 is not in the domain of , nosotros needed to consider the monotonicity of the function on simply the intervals and . The function is therefore increasing on its entire domain.

Therefore, our answer is option A; the function is increasing on and .

We volition now consider the criteria for an exponential office that would go far increasing.

Example 4: Identifying the Condition for an Exponential Function to be Increasing

What condition must there exist on for , where is a positive number, to exist an increasing function?

Answer

A office is increasing on an interval if for any in .

To ensure that our office, , is increasing for all positive values of , we need to recognize that we have an exponential function. The general form of an exponential function is . If , the function is increasing, and if , the role is decreasing. This can be partly shown as follows.

Starting with , we demand to notice the values of such that .

We can write

Nosotros can utilize this to compare the sizes of and

We know and , so

Nosotros likewise know , , so

For an increasing part, nosotros need the inequality to hold for all of these possible values of and . Substituting in the expression for , we can rewrite this inequality as

Since and are positive, this inequality will only be true if , for any positive value of . We can find the values of that satisfy this inequality by taking logarithms of both sides:

Since is positive, nosotros must have which is true when .

Hence, the function is increasing, for positive , when . In this question, so we have the inequality

Therefore, if , then is an increasing function for positive values of .

In our final example, we will consider the increasing and decreasing regions of a reciprocal function without beingness given its graph.

Example 5: Identifying the Increasing and Decreasing Regions of a Reciprocal Part

Which of the following statements is true for the function ?

is increasing on the intervals and .
is increasing on the intervals and .
is decreasing on the intervals and .
is decreasing on the intervals and .

Answer

We see that our function is a reciprocal function. Nosotros tin can notice the increasing and decreasing regions of a function from its graph, so 1 fashion of answering this question is to sketch the bend, .

Nosotros brainstorm by sketching the graph, . This graph has horizontal and vertical asymptotes fabricated upwardly of the - and -axes.

We will now consider the series of transformations that map the function onto .

Firstly, the graph of is a reflection of in the -axis and has the same horizontal and vertical asymptotes.

Next, nosotros tin can map onto by translating the graph 7 units correct. Since , which ways that the vertical asymptote is at present the line with equation , and since a horizontal translation does non touch on the position of the horizontal asymptote, this remains as the -axis.

The office contains the negative of this, so we can map onto by reflecting it in the -axis. The vertical and horizontal asymptotes are unchanged nether this transformation, as is the horizontal asymptote.

Finally, to map onto , we perform a vertical translation 5 units downward. This translates the horizontal asymptote down 5 units and leaves the vertical asymptote unchanged. Then, we now take a horizontal asymptote at .

The function is a reciprocal function with a vertical asymptote at and a horizontal asymptote at .

We now need to establish where this function is increasing and where it is decreasing.

We think that a role is increasing on an interval if for any in . A function is decreasing on an interval if for any in .

As we motility from to 7 forth the -axis, the output values of are decreasing.

This means that on the interval , the part is decreasing.

Too, as we motion from 7 to along the -axis, the output values of are decreasing. This ways that on the interval , the function is also decreasing.

It is important to note what is happening at . 7 is not in the domain of , which means that information technology cannot be in the intervals where is increasing or decreasing.

We tin can conclude that the function is decreasing on the intervals and ; in other words, information technology is a decreasing function.

Therefore, our answer is pick D; is decreasing on the intervals and .

We will finish this explainer by recapping some of the key points.

Key Points

A office is increasing on an interval if for any in .
A role is decreasing on an interval if for any in .
A role is constant on an interval if for any in , for some abiding .
A part tin exist increasing, decreasing, or constant for different intervals on its domain. We can identify these different regions from the graph of the function.
Alternatively, we describe a function as simply increasing, decreasing, or a constant, if this is true for its entire domain.