Determine the increasing and decreasing domain of the following function: The function describes a student's grades throughout the year.

Partly increasing and partly decreasing.

Which domain corresponds to the described function: The function represents the velocity of a stone after being dropped from a great height as a function of time.

Partly increasing and partly decreasing

Determine the domain of the following function: A function describing the charging of a computer battery during use.

Partly increasing and partly decreasing

Determine which domain corresponds to the described function: The function describes a person's energy level throughout the day.

Partly increasing and partly decreasing

Decreasing function

What is a Decreasing Function?

A decreasing function is a type of relationship where, as you move to the right on the graph (increasing the xxx-value), the $y$ -value gets smaller. It’s like going downhill—the farther you go (the more you increase $x$ ), the lower your height (the $y$ -value) becomes.

We will say that a function is decreasing when, as the value of the independent variable $X$ increases, the value of the function $Y$ decreases.

How to Spot a Decreasing Function:

On a Graph: The line or curve goes downward as you move from left to right.
In Numbers: For any two $x$ -values, if the second number is larger than the first $x_2 > x_1$, then the second $y$ -value will be smaller than the first $f(x_2) < f(x_1)$ .

Real-Life Example:

Think about eating a stack of cookies. Every time you eat one, the number of cookies left in the stack gets smaller. That’s a decreasing function—your $y$ -value (cookies left) decreases as your $x$ -value (number of cookies eaten) increases.

Fun Fact:

If the line or curve always goes down without stopping, it's called strictly decreasing. If it flattens for a bit before going down again, it’s just decreasing.

Let's see an example of strictly decreasing linear function on a graph:

Detailed explanation

Start practice

Test yourself on increasing and decreasing intervals of a function!

Does the function in the graph decrease throughout?

Practice more now

Let's assume we have two elements $X$ , which we will call $X1$ and $X2$ , where the following is true: X1<X2, meaning, X2 is located to the right of X1.

When $X1$ is placed in the domain, the value $Y1$ is obtained.
When $X2$ is placed in the domain, the value $Y2$ is obtained.

The function is decreasing when: $X2>X1$ and also $Y2<Y1$ .

The function can be decreasing in intervals or throughout its domain.

If you are interested in this article, you might also be interested in the following articles:

Graphical representation of a function

Algebraic representation of a function

Notation of a function

Domain of a function

Indefinite integral

Assignment of numerical value in a function

Variation of a function

Increasing function

Constant function

Functions for seventh grade

Intervals of increase and decrease of a function

In the blog of Tutorela you will find a variety of articles with interesting explanations about mathematics

Decreasing Function Exercises

Exercise 1

Assignment

Find the decreasing area of the function

$y=(x+1)+1$

Solution

$a$ coefficient of $x^2$

Therefore $0<a$

is the minimum point

The vertex of the function is $\left(-1,1\right)$

The function decreases in the area of $x<-1$

Answer

$x<-1$

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Test your knowledge

Question 1

Is the function in the graph decreasing?

Question 2

Is the function shown in the graph below decreasing?

Question 3

Is the function in the graph decreasing?

Exercise 2

Assignment

Given the function in the graph

When is the function positive?

Solution

The intersection point with the axis : $x$ is: $\left(-4,0\right)$

First positive, then negative.

Therefore $x<-4$

Answer

$x<-4$

Exercise 3

Assignment

Given the function in the diagram, what is its domain of positivity?

Solution

Note that the entire function is always above the axis: $x$

Therefore, it will always be positive. Its area of positivity will be for all $x$

Answer

For all $x$

Do you know what the answer is?

Question 1

Is the function in the graph below decreasing?

Question 2

Is the function shown in the graph below decreasing?

Question 3

In what domain does the function increase?

Exercise 4

Assignment

Given the function in the diagram

What are the areas of positivity and negativity of the function?

Solution

Let's remember that a function is positive when it is above the axis: $x$ and the function is negative when it is below the axis $x$

Given that the point of intersection with the axis: $x$ is $\left(3.5,0\right)$

When $x>3.5$ it is below: $x$

When $x<3.5$ it is above: $x$

Therefore, the function is positive when $x<3.5$ and negative when $x>3.5$

Answer

Positive when $x<3.5$

Negative when $x>3.5$

Exercise 5

Assignment

Find the increasing and decreasing area of the function

$f(x)=-2x^2+10$

Solution

In the first step, let's consider that $a=-2$

Therefore $x<0$ and the parabola is at its maximum

In the second step, find $x$ of the vertex

according to the data we know

$a=-2,b=0,c=10$

We replace the data in the formula

$x=\frac{-b}{2\cdot a}$

$x=\frac{-0}{2\cdot\left(-2\right)}$

$x=\frac{-0}{-4}$

$x=0$

Then we know that: $x=0$ and we replace it in the function and find that $y$

$y=10$

Answer

$0<x$ Decreasing

$x<0$ Increasing

Check your understanding

Question 1

Determine in which domain the function is negative?

Question 2

In what domain is the function increasing?

Question 3

In what domain does the function increase?

Examples with solutions for Decreasing function

Exercise #1

Is the function shown in the graph below decreasing?

Step-by-Step Solution

The graph presented is a straight line. To determine whether the function is decreasing, we need to examine the slope of this line.

The line has a negative slope, as it moves downward from left to right. A function is considered decreasing when its slope is negative.

In formal terms, for a linear function expressed as $y = mx + c$ , if the slope $m$ is negative, the function is decreasing over its entire domain.

From the graph, it's evident that the line has a negative slope, thus indicating that the function is indeed decreasing.

Therefore, the answer to the problem is Yes.

Answer

Yes

Exercise #2

Is the function in the graph decreasing?

Step-by-Step Solution

To analyze whether the function in the graph is decreasing, we must understand how the function's behavior is defined by its graph:

Step 1: Examine the graph. The graph presented is a horizontal line.
Step 2: Recognize the properties of a horizontal line. Horizontally aligned lines correspond to constant functions because the $y$ -value remains the same for all $x$ -values.
Step 3: Define the criteria for a function to be decreasing. A function decreases when, as $x$ increases, the value of $f(x)$ decreases.
Step 4: Apply this criterion to the horizontal line. Since the $y$ -value is constant and does not decrease as $x$ moves rightward, the function is not decreasing.

Therefore, the function represented by the graph is not decreasing.

Answer

Exercise #3

Is the function in the graph below decreasing?

Step-by-Step Solution

To determine if the function is decreasing, we will analyze the graph visually:

The graph shows a line connecting from the bottom-left to the top-right of the graph area, indicating the line has a positive slope. This type of graph indicates the function is increasing, not decreasing.

A decreasing function means its value goes down as $x$ increases, which is equivalent to having a negative slope.

Since the graph appears with a positive slope, the function is not decreasing.

Thus, the correct choice to the problem, which asks if the function in the graph is decreasing, is No.

Answer

Exercise #4

Does the function in the graph decrease throughout?

Step-by-Step Solution

To solve this problem, we'll begin by examining the graph of the function provided:

Step 1: Observe the graph from left to right along the x-axis.
Step 2: Look for any intervals where the function value (y-coordinate) does not decrease as the x-value increases.
Step 3: Pay special attention to segments where the graph might look horizontal or rising.

Upon inspecting the graph, we find:

- There are sections where the function's y-values appear to remain constant or potentially rise as the x-values increase. Specifically, even if the function decreases in major portions, any interval where it doesn't means the function cannot be classified as decreasing throughout.

Thus, the function does not strictly decrease on the entire interval shown. Therefore, the solution to the problem is No.

Answer

Exercise #5

Is the function in the graph decreasing?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Verify the graph's overall path direction
Step 2: Confirm if the y-values are decreasing as we proceed from the left side of the graph to the right side (increasing x-values).

Now, let's work through each step:

Step 1: By examining the graph, the red line starts at a higher point on the y-axis and moves downward to a lower point as it moves horizontally across the x-axis from left to right.

Step 2: Since for every point, the red line descends as it progresses from the leftmost point to the rightmost, this indicates a consistent decrease in the y-values.

Therefore, the solution to the problem is Yes, the function in the graph is decreasing.

Answer

Yes

Start practice

Decreasing function

Decreasing function

What is a Decreasing Function?

How to Spot a Decreasing Function:

Real-Life Example:

Fun Fact:

Test yourself on increasing and decreasing intervals of a function!

Decreasing Function Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Examples with solutions for Decreasing function

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Step-by-Step Solution

Answer

More Questions

Increasing and Decreasing Intervals of a Function