Decreasing function

🏆Practice increasing and decreasing intervals of a function

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

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Test yourself on increasing and decreasing intervals of a function!

einstein

In what interval is the function increasing?

Purple line: \( x=0.6 \)

111222333111000

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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, meaning, X2 is located to the right of 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 decreasing when: X2>X1 X2>X1 and also Y2<Y1 Y2<Y1 .

The function can be decreasing in intervals or throughout its domain.


Decreasing Function Exercises

Exercise 1

Assignment

Find the decreasing area of the function

y=(x+1)+1 y=(x+1)+1

Solution

a a coefficient of x2 x^2

Therefore 0<a 0<a

is the minimum point

The vertex of the function is (1,1) \left(-1,1\right)

The function decreases in the area of x<1 x<-1

Answer

x<1 x<-1


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Exercise 2

Assignment

Given the function in the graph

When is the function positive?

When is the function positive

Solution

The intersection point with the axis :x x is: (4,0) \left(-4,0\right)

First positive, then negative.

Therefore x<4 x<-4

Answer

x<4 x<-4


Exercise 3

Assignment

Given the function in the diagram, what is its domain of positivity?

Given the function in the diagram - what is its domain of positivity

Solution

Note that the entire function is always above the axis: x x

Therefore, it will always be positive. Its area of positivity will be for all x x

Answer

For all x x


Do you know what the answer is?

Exercise 4

Assignment

Given the function in the diagram

What are the areas of positivity and negativity of the function?

What are the areas of positivity and negativity of the function

Solution

Let's remember that a function is positive when it is above the axis: x x and the function is negative when it is below the axis x x

Given that the point of intersection with the axis: x x is (3.5,0) \left(3.5,0\right)

When x>3.5 x>3.5 it is below: x x

When x<3.5 x<3.5 it is above: x x

Therefore, the function is positive when x<3.5 x<3.5 and negative when x>3.5 x>3.5

Answer

Positive when x<3.5 x<3.5

Negative when x>3.5 x>3.5


Exercise 5

Assignment

Find the increasing and decreasing area of the function

f(x)=2x2+10 f(x)=-2x^2+10

Solution

In the first step, let's consider that a=2 a=-2

Thereforex<0 x<0 and the parabola is at its maximum

In the second step, find x x of the vertex

according to the data we know

a=2,b=0,c=10 a=-2,b=0,c=10

We replace the data in the formula

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

x=02(2) x=\frac{-0}{2\cdot\left(-2\right)}

x=04 x=\frac{-0}{-4}

x=0 x=0

Then we know that: x=0 x=0 and we replace it in the function and find that y y

y=10 y=10

Answer

0<x 0<x Decreasing

x<0 x<0 Increasing


Check your understanding

examples with solutions for decreasing function

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

Weather101010Speed

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

WeatherTemperature'000

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) (-1) .

Video Solution

Step-by-Step Solution

The function is:

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

Let's start by assuming that x equals 0:

f(0)=(1)×0=0 f(0)=(-1)\times0=0

Now let's assume that x equals minus 1:

f(1)=(1)×(1)=1 f(-1)=(-1)\times(-1)=1

Now let's assume that x equals 1:

f(1)=(1)×1=1 f(1)=(-1)\times1=-1

Now let's assume that x equals 2:

f(2)=(1)×2=2 f(2)=(-1)\times2=-2

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

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–3–3–3–2–2–2–1–1–1111222000

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:

Each number is divided by (1) (-1) .

Video Solution

Step-by-Step Solution

The function is:

f(x)=x1 f(x)=\frac{x}{-1}

Let's start by assuming that x equals 0:

f(0)=01=0 f(0)=\frac{0}{-1}=0

Now let's assume that x equals 1:

f(1)=11=1 f(1)=\frac{1}{-1}=-1

Now let's assume that x equals 2:

f(1)=11=1 f(-1)=\frac{-1}{-1}=1

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

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666–1–1–1111222333444000

We see that we got a decreasing function.

Answer

Decreasing

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