A relationship in which one value determines exactly one value of another quantity is called a function. In the first year of middle school, we study direct proportion and inverse proportion.
What Is a Function?
When the value of y is uniquely determined by choosing the value of x, we say that “y is a function of x.”
- Example: When buying x snacks at 80 yen each, the price y yen → y = 80x (a function)
- Example: For a person who is x cm tall, the weight y kg → Even people of the same height may have various weights (not a function)
The Coordinate Plane and Quadrants
The plane in which the x-axis and y-axis intersect at right angles is called the coordinate plane. The position of a point is represented by an ordered pair (x-coordinate, y-coordinate). The coordinate plane is divided into the first through fourth quadrants in counterclockwise order from the upper right.
In which quadrants are point A(3, -2) and point B(-4, 5)?
Solution: A has x > 0 and y < 0, so it is in the fourth quadrant. B has x < 0 and y > 0, so it is in the second quadrant.
Range
The range of possible values of a variable is called its range. It is expressed using inequalities. Use ≤ for an included endpoint and < for an endpoint that is not included.
Express “x is at least -1 and less than 4” as an inequality.
Solution: -1 ≤ x < 4
- Function: a relationship in which y is uniquely determined when x is fixed
- Coordinates are represented by an ordered pair (x, y). The origin is (0, 0).
- Range: use ≤ for an included endpoint and < for an endpoint that is not included.
Practice Problems
- Can the perimeter y cm of a square with side length x cm be called a function of x? Give a reason.
- In which quadrant, or on which axis, are points A(5, 2), B(-3, -1), and C(0, 4)?
- Express “x is greater than -2 and at most 5” as an inequality.
Answers and Explanations
- Answer: It is a function (y = 4x, so y is uniquely determined when x is fixed).
- Answer: A is in the first quadrant, B is in the third quadrant, and C is on the y-axis (it does not belong to a quadrant).
- Answer: -2 < x ≤ 5.