Determine which domain corresponds to the function described below: The function represents the height of a child from birth to first grade.

Partly increasing and partly decreasing.

Determine which domain corresponds to the function described below: The function represents the amount of fuel in a car's tank according to the distance traveled by the car.

Partly increasing and partly decreasing

Determine the domain of the function described below: The function represents the amount of water in a pool while it is being filled.

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

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 $x$ -value), the $y$ -value also gets bigger. It’s like climbing a hill—the higher you go (the more you increase $x$ ), the more your height (the $y$ -value) increases.

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

How to Spot an Increasing Function:

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

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 $y$ -values, and the number of days is the $x$ -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:

Detailed explanation

Start practice

Test yourself on increasing and decreasing intervals of a function!

Is the function in the graph decreasing?

Practice more now

For example
let's assume we have two elements $X$ , which we will call $X1$ and $X2$ , where the following is true: $X1<X2$ , that is, $X2$ 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 increasing when $X2>X1$ and also $Y2>Y1$ .
The function can be increasing in intervals or can be continuous throughout its domain.

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

$y=-(x+3)^2$

Solution

Solve the equation using the shortcut multiplication formula

$y=-x^2-6x-9$

From this, the data we have are:

$a=-1,b=-6,c=9$

Find the vertex using the formula

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

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

$x=\frac{6}{-2}$

$x=-3$

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

From this we know that: $a<0$

And therefore the function is maximum

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

Answer

$x<-3$

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?

Question 2

Is the function in the graph decreasing?

Question 3

Does the function in the graph decrease throughout?

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: $x$

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

According to the graph the function is positive

therefore $x>2$

Answer

$x>2$

Exercise 3

Assignment

Find the increasing area of the function

$y=-(x-6)^2$

Solution

Solve the equation using the shortcut multiplication formula

$y=-x^2+12x-36$

From this, the data we have are:

$a=-1,b=12,c=36$

Find the vertex by the formula

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

$x=\frac{-12}{2\cdot\left(-1\right)}$

$x=\frac{-12}{-2}$

$x=6$

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

From this we know that: $a<0$

Therefore the function is maximum

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

Answer

$6<x$

Do you know what the answer is?

Question 1

Is the function shown in the graph below decreasing?

Question 2

Is the function shown in the graph below decreasing?

Question 3

In what interval is the function increasing?

Purple line: $$ x=0.6 $$

Exercise 4

Assignment

Find the increasing area of the function

$y=-(2x+6)^2$

Solution

Solve the equation using the shortcut multiplication formula

$y=-4x^2-24x-36$

From this, the data we have are:

$a=-4,b=-24,c=-36$

Find the vertex using the formula

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

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

$x=\frac{24}{-8}$

$x=-3$

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

From this we know that $a<0$

Therefore, the function is maximum

The function is increasing from $-3<x$

Answer

$-3<x$

Exercise 5

Assignment

Find the increasing area of the function

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

Solution

Solve the equation using the shortcut multiplication formula

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

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

From this, the data we have are:

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

Find the vertex by the formula

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

$x=\frac{-6}{2\cdot3}$

$x=\frac{-6}{6}$

$x=-1$

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

$y=3\cdot1-6+9$

$y=3-6+9$

$y=6$

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

From this we know that: $a>0$

Therefore, the function is minimum

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

Answer

$-1<x$

Check your understanding

Question 1

In what domain does the function increase?

Question 2

Determine in which domain the function is negative?

Question 3

In what domain is the function increasing?

Examples with solutions for Increasing functions

Exercise #1

Is the function shown in the graph below decreasing?

Step-by-Step Solution

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

Step 1: Visually inspect the graph to see if it is consistently sloping downward.
Step 2: Apply the definition of a decreasing function.

Now, let's work through each step:
Step 1: Observing the graph, the function's graph is a line moving from the top left to the bottom right. This indicates it slopes downward as we move from left to right across the $x$ -axis.
Step 2: According to the definition of a decreasing function, for any $x_1 < x_2$ , it must hold true that $f(x_1) > f(x_2)$ . Since the graph shows a line moving downward, this condition is satisfied throughout its domain.

Therefore, the function represented by the graph is indeed decreasing.

The final answer 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

In what domain is the function increasing?

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 $x$

Exercise #4

In what interval is the function increasing?

Purple line: $x=0.6$

Video Solution

Step-by-Step Solution

Let's remember that a function is described as 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

$x<0.6$

Exercise #5

Determine in which domain the function is negative?

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 $x > 1$ the function is decreasing, meaning the Y values are decreasing.

Answer

$x > 1$

Start practice

Increasing functions

Increasing functions

What is an Increasing Function?

How to Spot an Increasing Function:

Real-Life Example:

Fun Fact:

Test yourself on increasing and decreasing intervals of a function!

Increasing Function

Increasing Function Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Examples with solutions for Increasing functions

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Increasing and Decreasing Intervals of a Function