# 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 $X$ increases, the value of the function $Y$ decreases.

## Test yourself on increasing and decreasing intervals of a function!

In what interval is the function increasing?

Purple line: $$x=0.6$$

Let's assume we have two elements $X$, which we will call $X1$ and $X2$, where the following is true: X1<X2, meaning, 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 decreasing when: $X2>X1$ and also $Y2.

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

## Decreasing Function Exercises

### Exercise 1

Assignment

Find the decreasing area of the function

$y=(x+1)+1$

Solution

$a$ coefficient of $x^2$

Therefore $0

is the minimum point

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

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

$x<-1$

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### Exercise 2

Assignment

Given the function in the graph

When is the function positive?

Solution

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

First positive, then negative.

Therefore $x<-4$

$x<-4$

### Exercise 3

Assignment

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

Solution

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

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

For all $x$

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?

Solution

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

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

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

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

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

Positive when $x<3.5$

Negative when $x>3.5$

### Exercise 5

Assignment

Find the increasing and decreasing area of the function

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

Solution

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

Therefore$x<0$ and the parabola is at its maximum

In the second step, find $x$ of the vertex

according to the data we know

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

We replace the data in the formula

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

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

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

$x=0$

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

$y=10$

$0 Decreasing

$x<0$ Increasing

## 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.

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.

### 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.

### 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$(-1)$.

### 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.

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 $(-1)$.

### Step-by-Step Solution

The function is:

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

Let's start by assuming that x equals 0:

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

Now let's assume that x equals 1:

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

Now let's assume that x equals 2:

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

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

We see that we got a decreasing function.