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.

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

What is a function?

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

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

An independent variable (X) (X) is a non-variable constant by which we explain (Y) (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 10 hours, the number of hours worked is actually the independent variable (X) (X) with which we know how much she earned. Ultimately this is the dependent variable. (Y) (Y)

In other words, it can be said that the amount Daniela earned is a function of the number of hours she worked (X) (X) .
We will mark the data of the function algebraically in this way: fx=X×30 fx=X\times30

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


Start practice

Test yourself on functions!

einstein

In what interval is the function increasing?

Purple line: \( x=0.6 \)

111222333111000

Practice more now

Now, the mathematical explanation of a function

Let's suppose we have two different groups in front of us, a first group and a second group, and each group has elements that belong exclusively to that same group. A function is actually our ability to pair each member of the first group with a unique member of the second group.

  • The first group includes elements called "variables"
  • While the second group includes the "function values" obtained for these "variables".

As we have already mentioned, for each variable there is a single function value, but for a specific function value, there can be several variables.

Variable ---------------------> Unique function value


Function Notation

Graphing a function is really about how the function is written. Usually, the variable (that is, the value that can be placed into a function) is denoted by x x or any other letter of the alphabet, while the value of the function for that variable x x is denoted by f(x) f\left(x\right) .


Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge

Representation of a Function

There are several ways to represent a function. We'll briefly mention them:

It's important to understand that each function can be represented in the ways described above, and an important part of understanding the topic of functions is the ability to "convert" one representation into another.

A1 - Representations of a function


Types of Functions

As mentioned, the topic of functions is a very broad subject and is taught from seventh to twelfth grade at various levels and within different subject frameworks.

  • Linear function
  • Quadratic function
  • Polynomial function
  • Rational function
  • Root of a function
  • Trigonometric function
  • Exponential function
  • Logarithmic function
  • Function with parameters
  • Even functions
  • Odd functions
  • And more...

Do you know what the answer is?

Characteristics of a Function

It's common to analyze functions according to the following sections:

  • Domain of a function: the values x x that can be input into a function (for a detailed explanation of "Domain of a function"). There are also functions that are not defined for certain domains or values (see the article "Undefined Function (Indefinite Integral)").
  • Intercepts with the axes - The common points of the function with the coordinate system.
  • Extreme points of a function: the points at which the function changes from increasing to decreasing and from decreasing to increasing.
  • Slope of a function: the rate at which a function changes (see the article "Equation with variable in the denominator").
  • Areas of increasing and decreasing of the function: the areas x x where the function increases or decreases (see the article "Areas of increase and decrease of the function").
A3 characteristics of a function


The function can also be

  • Constant function: the function values do not change for all values of x x
  • Increasing function: the function values increase as the values of x x increase
  • Decreasing function: the function values decrease as the values of x x increase

Check your understanding

Plugging a Numerical Value into a Function

We can place different numbers in place of the x x .

For example, if we have the function

f(x)=x+2 f(x)=x+2

We can substitute any number we want for x x . For each number we substitute, we get a different function value.

Let's look at some examples:

  • f(2)=2+2=4 f(2)=2+2=4
  • f(5)=5+2=7 f(5)=5+2=7
  • f(10)=10+2=12 f(10)=10+2=12
  • f(100)=100+2=102 f(100)=100+2=102
  • f(5)=5+2=3 f(-5)=-5+2=-3

Practice Examples and Functions for Seventh Grade

Exercise No. 1

Given the function Y=X+5 Y=X+5

A. What type of function is it?

B. Is the rate of change (slope) of the function constant? Also, what is the value of the slope?

C. Draw the graph of the function  

Solution: 

A. After a quick look at the function, we can determine that the function is linear. This is because it is the first power of X X .

B. The rate of change, that is, the slope of a linear function is constant and equal to the coefficient of X X . In our case, the coefficient of X X is equal to 1 1 . Therefore, the slope of the function is also equal to 1 1 .

C. To graph a linear function, only 2 2 points may be sufficient. We will add a third point to test ourselves.

For X=0 X=0 we get Y=5 Y=5

For X=1 X=1 we get Y=6 Y=6

For X=2 X=2 we get Y=7 Y=7

Now we will mark the points on the coordinate system and connect them:

Answer:

A. Linear function

B. Slope equal to 1 1 .

Do you think you will be able to solve it?

Exercise No. 2

Given the function F(x)=5X+3 F(x)=5X+3

How much is the function worth for the following values of X X ?

  • 0 0
  • 1 1
  • 2 2
  • 1 -1
  • 2 -2
  • 3 -3
  • X4 X-4
  • 5 5
  • 6 6
  • 7 7

Solution:

We will substitute the values we have in front of us for the X X in the function and obtain:

  • f(0)=5x+3=50+3=0+3=3 f(0)=5x+3=5\cdot0+3=0+3=3
  • f(2)=5x+3=52+3=10+3=13 f(2)=5x+3=5\cdot2+3=10+3=13
  • f(1)=5x+3=51+3=5+3=8 f(1)=5x+3=5\cdot1+3=5+3=8
  • f(1)=5x+3=5(1)+3=5+3=2 f(-1)=5x+3=5\cdot(-1)+3=-5+3=-2
  • f(2)=5x+3=5(2)+3=10+3=7 f(-2)=5x+3=5\cdot(-2)+3=-10+3=-7
  • f(3)=5x+3=5(3)+3=15+3=12 f(-3)=5x+3=5\cdot(-3)+3=-15+3=-12
  • f(X4)=5x+3=5(X4)+3=5X20+3=5X17 f(X-4)=5x+3=5\cdot(X-4)+3=5X-20+3=5X-17
  • f(5)=5x+3=55+3=25+3=28 f(5)=5x+3=5\cdot5+3=25+3=28
  • f(6)=5x+3=56+3=30+3=33 f(6)=5x+3=5\cdot6+3=30+3=33
  • f(7)=5x+3=57+3=35+3=38 f(7)=5x+3=5\cdot7+3=35+3=38

Examples and Exercises with Solutions for 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

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

Test your knowledge
Start practice