Increasing functions

Increasing functions - Examples, Exercises and Solutions
🏆Practice increasing and decreasing intervals of a function
5 Questions
Functions
Increasing and Decreasing Intervals of a Function
Practice
Add a new subject

Increasing functions

What is an Increasing Function?

An increasing function is a type of relationship where, as you move to the right on the graph (increasing the xx-value), the yy-value also gets bigger. It’s like climbing a hill—the higher you go (the more you increase xx), the more your height (the yy-value) increases.

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

How to Spot an Increasing Function:

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

Real-Life Example:

Think about saving money in a piggy bank. Every day you add more coins, and the total amount of money keeps going up. That’s an increasing function in action—your savings are the yy-values, and the number of days is the xx-values.

Fun Fact:

If the line or curve never stops going up, it's called strictly increasing. If it flattens for a bit before going up again, it's just increasing.

let's see an example of strictly increasing linear function:

increasing function

Start practice

Test yourself on increasing and decreasing intervals of a function!

Does the function in the graph decrease throughout?

YYYXXX

Practice more now

For example
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 X1<X2 , that is, X2 X2 is located to the right of X1 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 increasing when X2>X1 X2>X1 and also Y2>Y1 Y2>Y1 .
The function can be increasing in intervals or can be continuous throughout its domain. 

Increasing Function

increasing function


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

Decreasing 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

Increasing Function Exercises

Exercise 1

Assignment

Find the increasing area of the function

y=−(x+3)2 y=-(x+3)^2

Solution

Solve the equation using the shortcut multiplication formula

y=−x2−6x−9 y=-x^2-6x-9

From this, the data we have are:

a=−1,b=−6,c=9 a=-1,b=-6,c=9

Find the vertex using the formula

x=−b2⋅a x=\frac{-b}{2\cdot a}

x=−(−6)2⋅(−1) x=\frac{-\left(-6\right)}{2\cdot\left(-1\right)}

x=6−2 x=\frac{6}{-2}

x=−3 x=-3

Vertex point is (−3,0) \left(-3,0\right)

From this we know that: a<0 a<0

And therefore the function is maximum

The function is increasing in the area of x<−3 x<-3

Answer

x<−3 x<-3

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge
Question 1

Is the function in the graph below decreasing?

yx

Question 2

Is the function in the graph decreasing?

yx

Question 3

Is the function in the graph decreasing? yx

Exercise 2

Assignment

Given the linear function in the graph.

When is the function positive?

The function is positive when it is above the x-axis

Solution

The function is positive when it is above the axis: x x

Intersection point with the axis: x x is (2,0) \left(2,0\right)

According to the graph the function is positive

therefore x>2 x>2

Answer

x>2 x>2

Exercise 3

Assignment

Find the increasing area of the function

y=−(x−6)2 y=-(x-6)^2

Solution

Solve the equation using the shortcut multiplication formula

y=−x2+12x−36 y=-x^2+12x-36

From this, the data we have are:

a=−1,b=12,c=36 a=-1,b=12,c=36

Find the vertex by the formula

x=−b2⋅a x=\frac{-b}{2\cdot a}

x=−122⋅(−1) x=\frac{-12}{2\cdot\left(-1\right)}

x=−12−2 x=\frac{-12}{-2}

x=6 x=6

The vertex point is (6,0) \left(6,0\right)

From this we know that: a<0 a<0

Therefore the function is maximum

The function is increasing in the area of 6<x 6<x

Answer

6<x 6<x

Do you know what the answer is?
Question 1

Is the function shown in the graph below decreasing?

yx

Question 2

Is the function shown in the graph below decreasing?

yx

Question 3

Determine in which domain the function is negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Exercise 4

Assignment

Find the increasing area of the function

y=−(2x+6)2 y=-(2x+6)^2

Solution

Solve the equation using the shortcut multiplication formula

y=−4x2−24x−36 y=-4x^2-24x-36

From this, the data we have are:

a=−4,b=−24,c=−36 a=-4,b=-24,c=-36

Find the vertex using the formula

x=−b2⋅a x=\frac{-b}{2\cdot a}

x=−(−24)2⋅(−4) x=\frac{-\left(-24\right)}{2\cdot\left(-4\right)}

x=24−8 x=\frac{24}{-8}

x=−3 x=-3

The vertex point (−3,0) \left(-3,0\right)

From this we know that a<0 a<0

Therefore, the function is maximum

The function is increasing from −3<x -3<x

Answer

−3<x -3<x

Exercise 5

Assignment

Find the increasing area of the function

y=(x+3)2+2x2 y=(x+3)^2+2x^2

Solution

Solve the equation using the shortcut multiplication formula

y=x2+6x+9+2x2 y=x^2+6x+9+2x^2

y=3x2+6x+9 y=3x^2+6x+9

From this, the data we have are:

a=3,b=6,c=9 a=3,b=6,c=9

Find the vertex by the formula

x=−b2⋅a x=\frac{-b}{2\cdot a}

x=−62⋅3 x=\frac{-6}{2\cdot3}

x=−66 x=\frac{-6}{6}

x=−1 x=-1

Now replace x=−1 x=-1 in the given function

y=3⋅1−6+9 y=3\cdot1-6+9

y=3−6+9 y=3-6+9

y=6 y=6

The vertex point is (−1,6) \left(-1,6\right)

From this we know that: a>0 a>0

Therefore, the function is minimum

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

Answer

−1<x -1<x

Check your understanding
Question 1

In what domain does the function increase?

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

Question 2

In what domain does the function increase?

000

Question 3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Examples with solutions for Increasing functions

Exercise #1

Is the function shown in the graph below decreasing?

yx

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 y = mx + c , if the slope m 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?

yx

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 y -value remains the same for all x x -values.
  • Step 3: Define the criteria for a function to be decreasing. A function decreases when, as x x increases, the value of f(x) f(x) decreases.
  • Step 4: Apply this criterion to the horizontal line. Since the y y -value is constant and does not decrease as x x moves rightward, the function is not decreasing.

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

Answer

No

Exercise #3

Is the function in the graph below decreasing?

yx

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 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

No

Exercise #4

Does the function in the graph decrease throughout?

YYYXXX

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

No

Exercise #5

Is the function in the graph decreasing? yx

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

More Questions

Related Subjects
👉  Click here  to practice