Lessons 25–26 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
25
Relations and mappings; the function
Textbook, Part 2: pp. 3–7
GoalDistinguish relations from functions, name the domain and range, compute function values and test whether a relation is a function.
Any set of points in the coordinate plane (or any set of ordered pairs) is called a relation; it is often given by an equation in x and y. A relation with no two different points having the same first coordinate is a function (mapping): each x has at most one y. The set of first coordinates (the x’s) is the domain and the set of second coordinates (the y’s) is the range. If a relation is given by an equation, it is a function when every x gives a single y; on a graph, every vertical line must meet the graph in at most one point. In y = f(x), the value f(a) is the value of the function at x = a; to find it, substitute a for x in the formula. The symbol f is the rule itself, while f(x) is the result of the rule at x.
Worked examples
Let f(x) = 3x² − x. Then f(−2) = 3 · (−2)² − (−2) = 3 · 4 + 2 = 14. Also f(a + 1) = 3(a + 1)² − (a + 1) = 3a² + 6a + 3 − a − 1 = 3a² + 5a + 2.
{(1, 4), (2, 4), (3, 7)} is a function (the first coordinates differ); its domain is {1, 2, 3} and range {4, 7}. {(1, 4), (1, 5)} is not a function: x = 1 has two y-values.
Class activity
In pairs build a relation of 4–5 ordered pairs for your partner, who decides whether it is a function and writes its domain and range.
Practice
1
What is a function? Say it in your own words.
A relation in which each x has at most one y.
2
If f(x) = 4x − 7, find f(5).
13
3
Which of the relations y = |x| and x = |y| is a function?
y = |x| is a function (each x has one y). x = |y| is not: x = 2 gives y = 2 and y = −2.
4
Why is the relation x = y² not a function?
For x = 9 we get y = 3 and y = −3: two different points with the same first coordinate.