Increasing functions

Increasing functions - Examples, Exercises and Solutions
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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

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Test yourself on increasing and decreasing intervals of a function!

Determine in which domain the function is negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

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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

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Test your knowledge
Question 1

Does the function in the graph decrease throughout?

YYYXXX

Question 2

In what domain does the function increase?

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

Question 3

In what domain does the function increase?

000

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

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

Question 2

Is the function in the graph below decreasing?

yx

Question 3

Is the function in the graph decreasing?

yx

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

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 Increasing functions

Exercise #1

Determine in which domain the function is negative?

–0.5–0.5–0.50.50.50.51111.51.51.5222000

Video Solution

Step-by-Step Solution

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 observe that in the domain x>1 x > 1 the function is decreasing, meaning the Y values are decreasing.

Answer

x>1 x > 1

Exercise #2

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 #3

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 x values and y y values increase simultaneously.

On the other hand, the function decreases if the x x values increase while the y y values decrease simultaneously.

In the given graph, we can see that the function increases in the domain where x>0 x > 0 ; in other words, where the y y values are increasing.

Answer

x>0 x > 0

Exercise #4

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 x < 0 , meaning the Y values are increasing.

Answer

x<0 x<0

Exercise #5

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

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