Functions for Seventh Grade

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

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 $f\left(x\right)$.

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 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...

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").

• 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

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:

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

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 $1$.

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:

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