Increasing functions

🏆Practice increasing and decreasing intervals of a function

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.

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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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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



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=x26x9 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=b2a x=\frac{-b}{2\cdot a}

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

x=62 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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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=(x6)2 y=-(x-6)^2

Solution

Solve the equation using the shortcut multiplication formula

y=x2+12x36 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=b2a x=\frac{-b}{2\cdot a}

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

x=122 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?

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=4x224x36 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=b2a x=\frac{-b}{2\cdot a}

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

x=248 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=b2a x=\frac{-b}{2\cdot a}

x=623 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=316+9 y=3\cdot1-6+9

y=36+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

Examples with solutions for Increasing functions

Exercise #1

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 #2

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 #3

In what domain is the function increasing?

–5–5–5555101010151515–5–5–5555000

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 shown, we can see that the function is increasing in every domain, therefore the function is increasing for all X.

Answer

Entirex x

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 meaning the Y values are increasing.

Answer

x<0

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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