Functions

Learn the basic concepts of functions. Deepen your understanding of variables, function definitions, and domains through concrete examples.

Letters like $x$ and $y$ above that can take various values are called variables .
When the value of one variable $x$ is determined, and the value of $y$ is uniquely determined in correspondence with it, we say that $y$ is a function of $x$ .
The range of values that a variable can take is called the domain .
The domain can be expressed using inequalities, such as $a \leq x \leq b$ .

Relationships Between Quantities That Change Together

Example 1

Let's consider making a box using a square with a side length of 16cm.

How to make the box

Cut out squares of the same size from the four corners of a square paper with a side length of 16cm
Fold the paper along the dotted lines, tape the overlapping edges together to make a box without a lid

Let $x$ cm be the side length of the square to be cut out, and $y$ cm be the side length of the bottom of the box. Try moving the slider in the figure below to change the value of $x$ . You'll see that $y$ changes along with $x$ and can take various values.

Let's put the changes in $y$ as $x$ changes into a table and graph to capture their characteristics.

We can also express the relationship between $x$ and $y$ with an equation. Expressing it with an equation makes it very convenient to easily know how the value of $y$ changes when the value of $x$ changes.

y=-2x+16

Example 2

Next, in the box-making in Example 1, let $y$ cm² be the area of the bottom of the box. How does $y$ cm² (the area of the bottom of the box) change as the value of $x$ changes?

At this time, the pattern of changes in $x$ and $y$ can be shown in a table and graph like this. What is the correspondence relationship between $x$ and $y$ ?

The relationship between $x$ and $y$ can be expressed by this equation:

y=(-2x+16)^2

This is called a quadratic function, but quadratic functions are studied in the 9th grade, so you don't need to remember this term at the 7th-grade level.

Letters like $x$ and $y$ above that can take various values are called variables .
When the value of one variable $x$ is determined, and the value of $y$ is uniquely determined in correspondence with it, we say that $y$ is a function of $x$ .

Range of Values a Variable Can Take

Example 3

The following table shows data on temperature and the number of cold drinks sold at a certain store in a certain country. What happens to the number of cold drinks sold when the temperature rises? Or conversely, what happens when the temperature drops?

When we plot this data on a graph, it seems to form a straight line, and the relationship between $x$ and $y$ can be expressed by the equation $y=5x+10$ .

However, $x$ here represents temperature, so is a value like $x=50\,(^\circ C)$ possible...? If the historical lowest temperature in the area where this data was taken is $10\,(^\circ C)$ and the highest is $40\,(^\circ C)$ , it's reasonable to consider that the range of values $x$ can take is from $10^\circ C$ to $40^\circ C$ . This range of values that a variable can take is called the domain .

The range of values that a variable can take is called the domain .
The domain can be expressed using inequalities, such as $a \leq x \leq b$ .