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Functions

Increasing and Decreasing Intervals of a Function

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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)

$X$ 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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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.

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").

Check your understanding

Question 1

Determine which domain corresponds to the function described below:

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

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: