## 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 the data is that Romina worked as a babysitter and earned 30 pesos 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 Romina worked, she received both $300$ pesos and $200$ pesos.

## 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**

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## 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 $x$ or any other letter of the alphabet, while the value of the function for that variable $x$ is denoted by $f\left(x\right)$.

## Representation of a Function

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

- Verbal representation of a function and tabular representation of a function (for complete reading "Verbal and tabular representation of a function ")
- Graphical representation of a function (see the article "Graphical representation of a function ")
- Algebraic representation of a function (see the article "Algebraic representation of a function ").

It's important to understand that each function can be represented in the 4 ways described above, and an important part of understanding the topic of functions is the ability to "convert" one representation into another.

## 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...

## Characteristics of a Function

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

**Domain of a function**: the values $x$ 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 $x$ where the function increases or decreases (see the article "Areas of increase and decrease of the function").

## The function can also be

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

## Plugging a Numerical Value into a Function

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

For example, if we have the function

#### $f(x)=x+2$

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

**Let's look at some examples:**

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

## Practice Examples and Functions for Seventh Grade

### Exercise No. 1

**Given the function ****$Y=X+5$**

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 $X$.

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

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

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

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

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

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

**Answer:**

A. Linear function

B. Slope equal to $1$.

### Exercise No. 2

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

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

- $0$
- $1$
- $2$
- $-1$
- $-2$
- $-3$
- $X-4$
- $5$
- $6$
- $7$

**Solution:**

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

- $f(0)=5x+3=5\cdot0+3=0+3=3$
- $f(2)=5x+3=5\cdot2+3=10+3=13$
- $f(1)=5x+3=5\cdot1+3=5+3=8$
- $f(-1)=5x+3=5\cdot(-1)+3=-5+3=-2$
- $f(-2)=5x+3=5\cdot(-2)+3=-10+3=-7$
- $f(-3)=5x+3=5\cdot(-3)+3=-15+3=-12$
- $f(X-4)=5x+3=5\cdot(X-4)+3=5X-20+3=5X-17$
- $f(5)=5x+3=5\cdot5+3=25+3=28$
- $f(6)=5x+3=5\cdot6+3=30+3=33$
- $f(7)=5x+3=5\cdot7+3=35+3=38$

### Exercise No. 3

**Here are the following drawings:**

Determine for each drawing if it is an increasing, decreasing, or constant function and explain why.

**Solution:**

A. This is an increasing function because if we look from left to right, the function values increase as the values of $x$ increase.

B. This is a decreasing function because if we look from left to right, the function values decrease as the values of $x$ decrease.

- This is a constant function because if we look from left to right, the function's values do not change at all as the values of $x$ increase.

**Answer:**

A. Increasing Function

B. Decreasing Function

C. Constant Function