Lesson Explainer: Increasing and Decreasing Intervals of a Function Mathematics

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 notation to describe the intervals of increase and decrease. We begin by recalling what we mean by interval notation.

Definition: Interval Notation

The interval of numbers between 𝑎 and 𝑏, including 𝑎 and 𝑏, is often denoted by [𝑎,𝑏], where 𝑎 and 𝑏 are called the endpoints of the interval.

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can either be replaced with a parenthesis or reversed. Thus, in set-builder notation, this is expressed as follows: (𝑎,𝑏)=]𝑎,𝑏[={𝑥𝑎<𝑥<𝑏},[𝑎,𝑏)=[𝑎,𝑏[={𝑥𝑎𝑥<𝑏},(𝑎,𝑏]=]𝑎,𝑏]={𝑥𝑎<𝑥𝑏},[𝑎,𝑏]=[𝑎,𝑏]={𝑥𝑎𝑥𝑏}.

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

Definition: Increasing Functions

A function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥).

Since we need to compare 𝑓(𝑥) to 𝑓(𝑥), the function 𝑓 must be defined on ]𝑎,𝑏[. When a function 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 function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥).

When a function is decreasing on an interval, its outputs are decreasing on this interval, so its curve must be falling on this interval.

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

We can also describe functions that do not change outputs on an interval as follows.

Definition: Constant Function on an Interval

A function 𝑓(𝑥) is constant on an interval ]𝑎,𝑏[ if for any 𝑥 in ]𝑎,𝑏[: 𝑓(𝑥)=𝑐, for some constant 𝑐.

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

Definition: Increasing, Decreasing, or Constant Functions

If a function 𝑓(𝑥) is increasing on its entire domain, we just say the function is increasing. Likewise, if a function 𝑓(𝑥) is decreasing on its entire domain, we just say the function is decreasing. Finally, if a function 𝑓(𝑥) is constant on its entire domain, we just say the function is constant.

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

Example 1: Identifying Whether the Function 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?

  1. The function is constant on .
  2. The function is increasing on .
  3. The function is decreasing on .
  4. The function is constant on ],0].
  5. The function is increasing on ],0].

Answer

Let us begin by recalling what the words increasing, decreasing, and constant tell us about the graphs of a function.

Firstly, a function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥). Secondly, a function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥). Finally, a function 𝑓(𝑥) is constant on an interval ]𝑎,𝑏[ if for any 𝑥 in ]𝑎,𝑏[𝑓(𝑥)=𝑐, for some constant 𝑐.

If we compare these definitions to our graph, we see that we have a horizontal line, so our function must be constant. For each value of 𝑥, we see that the output, 𝑦, is equal to 4.

If we look at the horizontal line representing our function, we see that it has arrows at both ends. This means that the line must extend from to . The interval ],[ is the set of all real numbers; therefore, the function must be constant for all real numbers.

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

In our next example, we will use the graph of a function to determine the intervals over which it is increasing, decreasing, or constant.

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

Which of the following statements correctly describe the monotonicity of the function represented in the figure below?

  1. The function is increasing on ]5,8[ and decreasing on ]2,5[.
  2. The function is increasing on ]2,5[ and decreasing on ]5,8[.
  3. The function is increasing on ]2,1[, constant on ]1,5[, and decreasing on ]5,8[.
  4. The function is increasing on ]5,8[, constant on ]1,5[, and decreasing on ]2,1[.

Answer

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

We recall that a function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥). A function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥). Finally, a function 𝑓(𝑥) is constant on an interval ]𝑎,𝑏[ if for any 𝑥 in ]𝑎,𝑏[: 𝑓(𝑥)=𝑐, for some constant 𝑐.

Our graph has three main sections.

Between 𝑥=2 and 𝑥=1, as the value of 𝑥 increases, the output, 𝑦, also increases. This means that the function is increasing on ]2,1[. As the options use open intervals, we do not need to worry about the endpoints of the intervals in this question.

Between 𝑥=1 and 𝑥=5, the output, 𝑦, is always equal to 3, so the function is constant on this interval. This means that the function is constant on ]1,5[. Once again, our interval is open at both ends.

Between 𝑥=5 and 𝑥=8, as the value of 𝑥 increases, the output, 𝑦, decreases. This means that the function is decreasing on ]5,8[.

The monotonicity of the function can be described as increasing on ]2,1[, constant on ]1,5[, and decreasing on ]5,8[.

Therefore, our answer is option C; the function is increasing on ]2,1[, constant on ]1,5[, and decreasing on ]5,8[.

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

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

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

  1. The function is increasing on ],0[ and ]0,[.
  2. The function is decreasing on ],0[ and ]0,[.
  3. The function is increasing on ],5[ and ]5,[.
  4. The function is decreasing on ],5[ and ]5,[.

Answer

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 function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥). A function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥).

Our graph has two asymptotes. We see that the 𝑦-axis (𝑥=0) is a vertical asymptote and we have a horizontal asymptote at 𝑦=5. This means that 0 is not in the domain of 𝑓.

As 0 is not in the domain of 𝑓, we need to consider the monotonicity of the function on its domain, the intervals ],0[ and ]0,[.

Let us now consider what happens to our graph as 𝑥 increases. As we move from to 0 along the 𝑥-axis, the value of 𝑓(𝑥) increases. This means that on the interval ],0[, the function is increasing.

The same thing occurs as we move from 0 to along the 𝑥-axis; the outputs of 𝑓(𝑥) are increasing. This means that on the interval ]0,[, the function is also increasing.

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

We can conclude that the function is increasing on ],0[ and ]0,[.

As 0 is not in the domain of 𝑓, we needed to consider the monotonicity of the function on just the intervals ],0[ and ]0,[. The function is therefore increasing on its entire domain.

Therefore, our answer is option A; the function is increasing on ],0[ and ]0,[.

We will now consider the criteria for an exponential function that would make it increasing.

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

What condition must there be on 𝑧 for 𝑓(𝑥)=𝑧7, where 𝑥 is a positive number, to be an increasing function?

Answer

A function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥).

To ensure that our function, 𝑓(𝑥)=𝑧7, is increasing for all positive values of 𝑥, we need to recognize that we have an exponential function. The general form of an exponential function is 𝑓(𝑥)=𝑎. If 𝑎>1, the function is increasing, and if 0<𝑎<1, the function is decreasing. This can be partly shown as follows.

Starting with 0<𝑥<𝑥, we need to find the values of 𝑎 such that 𝑓(𝑥)<𝑓(𝑥).

We can write 𝑥=𝑥+𝑝,𝑝>0.where

We can use this to compare the sizes of 𝑓(𝑥) and 𝑓(𝑥)𝑓(𝑥)=𝑎=𝑎=𝑎×𝑎=𝑎𝑓(𝑥).

We know 𝑎>0 and 𝑝>0, so 𝑎>0.

We also know 𝑥, 𝑥>0, so 𝑓(𝑥),𝑓(𝑥)>0.

For an increasing function, we need the inequality 𝑓(𝑥)<𝑓(𝑥) to hold for all of these possible values of 𝑥 and 𝑥. Substituting in the expression for 𝑓(𝑥), we can rewrite this inequality as 𝑓(𝑥)<𝑎𝑓(𝑥).

Since 𝑓(𝑥) and 𝑎 are positive, this inequality will only be true if 𝑎>1, for any positive value of 𝑝. We can find the values of 𝑎 that satisfy this inequality by taking logarithms of both sides: logloglogloglog(𝑎)>(1),𝑝(𝑎)>(1),𝑝(𝑎)>0.

Since 𝑝 is positive, we must have log(𝑎)>0, which is true when 𝑎>1.

Hence, the function 𝑎 is increasing, for positive 𝑥, when 𝑎>1. In this question, 𝑎=𝑧7 so we have the inequality 𝑧7>1(7)𝑧>7.multiplyingbothsidesby

Therefore, if 𝑧>7, then 𝑓(𝑥)=𝑧7 is an increasing function for positive values of 𝑥.

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

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

Which of the following statements is true for the function (𝑥)=17𝑥5?

  1. (𝑥) is increasing on the intervals ],7[ and ]7,[.
  2. (𝑥) is increasing on the intervals ],7[ and ]7,[.
  3. (𝑥) is decreasing on the intervals ],7[ and ]7,[.
  4. (𝑥) is decreasing on the intervals ],7[ and ]7,[.

Answer

We see that our function is a reciprocal function. We can find the increasing and decreasing regions of a function from its graph, so one way of answering this question is to sketch the curve, (𝑥)=17𝑥5.

We begin by sketching the graph, 𝑓(𝑥)=1𝑥. This graph has horizontal and vertical asymptotes made up of the 𝑥- and 𝑦-axes.

We will now consider the series of transformations that map the function 𝑓(𝑥)=1𝑥 onto (𝑥).

Firstly, the graph of 1(𝑥) is a reflection of 1𝑥 in the 𝑦-axis and has the same horizontal and vertical asymptotes.

Next, we can map 1(𝑥) onto 17𝑥 by translating the graph 7 units right. Since 17𝑥=1(𝑥7), which means that the vertical asymptote is now the line with equation 𝑥=7, and since a horizontal translation does not affect the position of the horizontal asymptote, this remains as the 𝑥-axis.

The function (𝑥) contains the negative of this, so we can map 17𝑥 onto 17𝑥 by reflecting it in the 𝑥-axis. The vertical and horizontal asymptotes are unchanged under this transformation, as 𝑦=0 is the horizontal asymptote.

Finally, to map 17𝑥 onto (𝑥)=17𝑥5, we perform a vertical translation 5 units downward. This translates the horizontal asymptote down 5 units and leaves the vertical asymptote unchanged. So, we now have a horizontal asymptote at 𝑦=5.

The function (𝑥)=17𝑥5 is a reciprocal function with a vertical asymptote at 𝑥=7 and a horizontal asymptote at 𝑦=5.

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

We recall that a function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥). A function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥).

As we move from to 7 along the 𝑥-axis, the output values of (𝑥) are decreasing.

This means that on the interval ],7[, the function is decreasing.

Likewise, as we move from 7 to along the 𝑥-axis, the output values of (𝑥) are decreasing. This means that on the interval ]7,[, the function is also decreasing.

It is important to note what is happening at 𝑥=7. 7 is not in the domain of 𝑓(𝑥), which means that it cannot be in the intervals where 𝑓(𝑥) is increasing or decreasing.

We can conclude that the function (𝑥)=17𝑥5 is decreasing on the intervals ],7[ and ]7,[; in other words, it is a decreasing function.

Therefore, our answer is option D; (𝑥) is decreasing on the intervals ],7[ and ]7,[.

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

Key Points

  • A function 𝑓(𝑥) is increasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)<𝑓(𝑥).
  • A function 𝑓(𝑥) is decreasing on an interval ]𝑎,𝑏[ if for any 𝑥<𝑥 in ]𝑎,𝑏[𝑓(𝑥)>𝑓(𝑥).
  • A function 𝑓(𝑥) is constant on an interval ]𝑎,𝑏[ if for any 𝑥 in ]𝑎,𝑏[𝑓(𝑥)=𝑐, for some constant 𝑐.
  • A function can be 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.

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