Decreasing function

Decreasing function - Examples, Exercises and Solutions
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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 yy-value gets smaller. It’s like going downhill—the farther you go (the more you increase xx), the lower your height (the yy-value) becomes.

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

How to Spot a Decreasing Function:

  1. On a Graph: The line or curve goes downward as you move from left to right.
  2. In Numbers: For any two xx-values, if the second number is larger than the first \(x_2 > x_1\​), then the second yy-value will be smaller than the first f(x2)<f(x1)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 yy-value (cookies left) decreases as your xx-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:

Decreasing function

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Test yourself on increasing and decreasing intervals of a function!

einstein

Does the function in the graph decrease throughout?

YYYXXX

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

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

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

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

Decreasing function

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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 y=(x+1)+1

Solution

a a coefficient of x2 x^2

Therefore 0<a 0<a

is the minimum point

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

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

Answer

x<−1 x<-1

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Test your knowledge
Question 1

In what domain does the function increase?

000

Question 2

In what domain does the function increase?

–20–20–20–10–10–10101010202020–10–10–10101010000

Question 3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Exercise 2

Assignment

Given the function in the graph

When is the function positive?

When is the function positive

Solution

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

First positive, then negative.

Therefore x<−4 x<-4

Answer

x<−4 x<-4

Exercise 3

Assignment

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

Given the function in the diagram - what is its domain of positivity

Solution

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

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

Answer

For all x x

Do you know what the answer is?
Question 1

In what domain is the function negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Question 2

Is the function in the graph below decreasing?

yx

Question 3

Is the function in the graph decreasing?

yx

Exercise 4

Assignment

Given the function in the diagram

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

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 x and the function is negative when it is below the axis x x

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

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

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

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

Answer

Positive when x<3.5 x<3.5

Negative when x>3.5 x>3.5

Exercise 5

Assignment

Find the increasing and decreasing area of the function

f(x)=−2x2+10 f(x)=-2x^2+10

Solution

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

Thereforex<0 x<0 and the parabola is at its maximum

In the second step, find x x of the vertex

according to the data we know

a=−2,b=0,c=10 a=-2,b=0,c=10

We replace the data in the formula

x=−b2⋅a x=\frac{-b}{2\cdot a}

x=−02⋅(−2) x=\frac{-0}{2\cdot\left(-2\right)}

x=−0−4 x=\frac{-0}{-4}

x=0 x=0

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

y=10 y=10

Answer

0<x 0<x Decreasing

x<0 x<0 Increasing

Check your understanding
Question 1

Is the function in the graph decreasing? yx

Question 2

Is the function shown in the graph below decreasing?

yx

Question 3

Is the function shown in the graph below decreasing?

yx

Examples with solutions for Decreasing function

Exercise #1

In what domain does the function increase?

000

Video Solution

Step-by-Step Solution

Let's remember that the function increases if the X values and Y values increase simultaneously.

On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.

In the given graph, we notice that the function increases in the domain where x < 0 , meaning the Y values are increasing.

Answer

x<0

Exercise #2

In what domain does the function increase?

–20–20–20–10–10–10101010202020–10–10–10101010000

Video Solution

Step-by-Step Solution

Let's remember that the function increases if the X values and Y values increase simultaneously.

On the other hand, the function decreases if the X values increase and the Y values decrease simultaneously.

In the given graph, we notice that the function increases in the domain where x > 0 meaning the Y values are increasing.

Answer

x > 0

Exercise #3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Video Solution

Step-by-Step Solution

Let's first remember that a function is increasing if both the X and Y values are increasing simultaneously.

Conversely, a function is decreasing if the X values are increasing while the Y values are decreasing simultaneously.

In the graph shown, we can see that the function is increasing in every domain and therefore the function is increasing for all values of X.

Answer

All values of x x

Exercise #4

In what domain is the function negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Video Solution

Step-by-Step Solution

Let's remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can see that in the domain x > 1 the function is decreasing, meaning the Y values are decreasing.

Answer

x > 1

Exercise #5

In what interval is the function increasing?

Purple line: x=0.6 x=0.6

111222333111000

Video Solution

Step-by-Step Solution

Let's remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can see that in the domain x < 0.6 the function is increasing, meaning the Y values are increasing.

Answer

x<0.6

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