On one hand, functions are a fairly abstract concept, but on the other hand, they are very useful in many areas of mathematics. The topic of functions dominates many fields, including algebra, trigonometry, differential and integral calculus, and more. Therefore, it's important to understand the concept of functions, so that it can be applied in any of the fields of mathematics, and especially when we start learning about functions in seventh grade.

## What is a function?

A function expresses a relationship between two variables (X and Y)

• $X$ represents an independent variable
• $Y$ represents a dependent variable

An independent variable $(X)$ is a non-variable constant by which we explain $(Y)$, the dependent variable

For example, if Daniela worked as a babysitter and earned 30 dollars per hour and we want to know how much Daniela made after $10$ hours, the number of hours worked is actually the independent variable $(X)$ with which we know how much she earned. Ultimately this is the dependent variable. $(Y)$

In other words, it can be said that the amount Daniela earned is a function of the number of hours she worked $(X)$.
We will mark the data of the function algebraically in this way: $fx=X\times30$

It's important to remember that each element in the domain $X$ will always have only one element in the range $Y$.
This means that it's not possible that during the $10$ hours Daniela worked, she received both $300$ dollars and $200$ dollars.

## Examples with solutions for Functions

### Exercise #1

In what interval is the function increasing?

Purple line: $x=0.6$

### 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.

x<0.6

### Exercise #2

In what domain does the function increase?

### 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.

x > 0

### Exercise #3

In what domain is the function negative?

### 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.

x > 1

### Exercise #4

In what domain is the function increasing?

### 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.

Entire$x$

### Exercise #5

In what domain does the function increase?

### 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.

x<0 

### Exercise #6

Determine which domain corresponds to the function described below:

The function represents the height of a child from birth to first grade.

### Step-by-Step Solution

According to logic, a child's height from birth until first grade will always be increasing as the child grows.

Therefore, the domain that suits this function is - always increasing.

Always increasing.

### Exercise #7

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 #8

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.

### Step-by-Step Solution

According to logic, the speed of the stone during a fall from a great height will increase as it falls with acceleration.

In other words, the speed of the stone increases, so the appropriate domain for this function is - always increasing.

Always increasing

### Exercise #9

Determine the domain of the following function:

A function describing the charging of a computer battery during use.

### Step-by-Step Solution

According to logic, the computer's battery during use will always decrease since the battery serves as an energy source for the computer.

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

Always decreasing

### Exercise #10

Determine the domain of the following function:

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

### Step-by-Step Solution

According to logic, the student's grades throughout the year depend on many criteria that are not given to us.

Therefore, the appropriate domain for the function is - it is impossible to know.

Impossible to know.

### Exercise #11

Determine the domain of the following function:

The function represents the weight of a person over a period of 3 years.

### Step-by-Step Solution

Logically, a person's weight is something that fluctuates.

In one week, a person's weight can increase, but in the following week, it can decrease.

Therefore, the domain that suits this function is - partly increasing and partly decreasing.

Partly increasing and partly decreasing.

### Exercise #12

In what domain does the function increase?

Black line: $x=1.1$

### Step-by-Step Solution

Remember that a function is increasing if the X values and Y values are increasing simultaneously.

A function is decreasing if the X values are increasing and the Y values are decreasing simultaneously.

In the plotted graph, we can see that in the domain 1.1 > x > 0 the function is increasing, meaning the Y values are increasing.

1.1 > x > 0

### Exercise #13

In which interval does the function decrease?

Red line: $x=0.65$

### 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 plotted graph, we can see that the function is decreasing in all domains. In other words, it is decreasing for all X.

All values of $x$

### Exercise #14

In which field does the function decrease?

Green line $x=-0.45$

Blue line $x=0.45$

### 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 -0.45 < x < 0.45 meaning the Y values are increasing.

0.45 > x > -0.45

### Exercise #15

In what domain does the function increase?

### 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 1 > x > -1 meaning the Y values are increasing.