In what domain does the function increase? Green line: $$ x=-0.8 $$ –2 –2 –2 2 2 2 2 2 2 0 0 0

It does not have an increasing domain.

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

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

Partly increasing and partly decreasing.

Increasing functions

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.

Detailed explanation

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Test yourself on increasing and decreasing intervals of a function!

In what domain does the function increase?

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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

$x<-3$

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Test your knowledge

Question 1

In what domain is the function negative?

Question 2

In what domain is the function increasing?

Question 3

In what domain does the function increase?

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

In what interval is the function increasing?

Purple line: $$ x=0.6 $$

Question 2

In what domain does the function increase?

Green line:
$$ x=-0.8 $$

Question 3

In which interval does the function decrease?

Red line: $$ x=0.65 $$

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 which interval does the function decrease?

Red line: $$ x=1.3 $$

Question 2

In what domain does the function increase?

Black line: $$ x=1.1 $$

Question 3

Determine the domain of the following function:

The function describes a student's grades throughout the year.

examples with solutions for increasing functions

Exercise #1

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.

Step-by-Step Solution

According to the definition, the amount of fuel in the car's tank will always decrease, since during the trip the car consumes fuel in order to travel.

Therefore, the domain that is suitable for this function is - always decreasing.

Answer

Always decreasing

Exercise #2

Choose the graph that best describes the following:

The acceleration of a ball (Y) after throwing it from a building as a function of time (X).

Step-by-Step Solution

Since acceleration is dependent on time, it will be constant.

The force of gravity on Earth is constant, meaning the velocity of Earth's gravity is constant and therefore the graph will be straight.

The graph that appears in answer B satisfies this.

Answer

Exercise #3

Choose the graph that best represents the following:

Temperature of lukewarm water (Y) after placing in the freezer as a function of time (X).

Step-by-Step Solution

Since the freezing point of water is below 0, the temperature of the water must drop below 0.

The graph in answer B describes a decreasing function and therefore this is the correct answer.

Answer

Exercise #4

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

For each number, multiply by $(-1)$ .

Video Solution

Step-by-Step Solution

The function is:

$f(x)=(-1)x$

Let's start by assuming that x equals 0:

$f(0)=(-1)\times0=0$

Now let's assume that x equals minus 1:

$f(-1)=(-1)\times(-1)=1$

Now let's assume that x equals 1:

$f(1)=(-1)\times1=-1$

Now let's assume that x equals 2:

$f(2)=(-1)\times2=-2$

Let's plot all the points on the function graph:

We can see that the function we got is a decreasing function.

Answer

Decreasing

Exercise #5

Determine whether the function is increasing, decreasing, or constant. For each function check your answers with a graph or table.

For each number, multiply by 0.

Video Solution

Step-by-Step Solution

The function is:

$f(x)=x\times0$

Let's start by assuming that x equals 0:

$f(0)=0\times0=0$

Now let's assume that x equals 1:

$f(1)=1\times0=0$

Now let's assume that x equals -1:

$f(-1)=(-1)\times0=0$

Now let's assume that x equals 2:

$f(2)=2\times0=0$

Let's plot all the points on the function's graph:

We can see that the function we obtained is a constant function.

Answer

Constant

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