Function Notation

Example 1

Which of the following are functions?

f = { (1, 2), (3, 6), (4, -2), (8, 0), (9, 6)}

g = { (1, 8), (4, 2), (3, 5), (1, 3), (6, 11)}

h = { (3, 4), (5, 4), (6, 1), (7, 3), (9, 0), (10, -2)}

Solution .

f is a function since it gives a rule that assigns each member of the set {1, 3, 4, 8, 9} exactly one value in the set {2, 6, -2, 0, 6}. Stated another way, f is a function because it never has two ordered pairs with the same x and different y values.

g is not a function because it does not assign each member of {1, 4, 3, 6} to exactly one value in {8, 2, 5, 3, 11} since it assigns x =1 to both y = 8 and y = 3. In other words, g is not a function because it has two ordered pairs, namely (1, 8) and (1, 3), which have the same x value but different y values.

h is a function since it assigns the members of {3, 5, 6, 7, 9, 10} to exactly one value in {4, 1, 3, 0, -2}. There are no pairs with the same x and different y values.

Example 2

Which of the following equations represent y as a function of x?

a.) b.)

Solution .

a.) This can be written as Since for each value of x there is exactly one value of y, y is a function of x.

b.) Solving this for y we get y = +/- Because there are two values of y for each x, this equations does not give y as a function of x.

Example 3

For the function h = { (3, 4), (5, 4), (6, 1), (7, 3), (9, 0), (10, -2)}, find h(3), h(5), and h(9).

Solution .

Since h(3) is the y value corresponding to 3, h(3) = 4. In addition, h(5) = 4 and h(9) = 0.

Example 4

If , find

a.) f(2) b.) f(k) c.) f(x + 1)

Solution .

a.) f(2) = - 4 + 6 + 4 = 6

b.)

c.)

A function defined using two or more equations is called a piecewise-defined function. This type of function is illustrated in the next example.

Example 5

Define f(x) by the following

Find f(-2), f(-1), f(0), f(1), f(2), f(3), and f(4)

Solution .

When x is less than or equal to 1, f(x) squares x. Consequently,

f(-2) = 4, f( -1) = 1, f(0) = 0, and f(1) = 1.

When x>1 then f(x) is three times x. Because of this we find that

f(2) = 6, f( 3) = 9, and f(4) = 12.