Chapter 4.5

relation

A set of any ordered pairs.

function

A special kind of relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

writing an equation using function notation

Step 1: Solve the equation for y. Step 2: Replace y with f(x).

relations and functions can be represented by

tables and graphs

the range of a nonconstant linear function

(-∞, ∞)

f(x)

Another name for the dependent variable y.

relations and functions can be expressed as

correspondence or mapping

the domain of any linear function

(-∞, ∞)

linear function

A function that can be defined by f(x) = ax + b, for real numbers a and b.

constant function

A linear function defined by f(x) = b (whose graph is a horizontal line)

range

the set of all values of the dependent variable (y)

domain

the set of all values of the independent variable (x)

vertical line test

If every vertical line intersects the graph of the relation in no more than one point, then the relation represents a function.

agreement on domain

The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable (cannot be numbers that lead to division by 0 or an even root of a negative number).

sometimes

letters other than f, such as g, h, or capital F, G, and H are used to name functions.

identifying functions from their equations (e)

y = 5/ x - 1 Defines a function. Domain includes all real numbers except those that make the denominator 0. x - 1 = 0, solve for x. The domain is (-∞, 1) ∪ (1, ∞)

function notation

y = f(x)

identifying functions from their equations (a)

y = x + 4 y is always found by adding 4 to x, therefore each value of x corresponds to just one value of y and the relation defines a function. The domain is (-∞,∞)

identifying functions from their equations (b)

y = √2x - 1 For any choice of x in domain, there is exactly one corresponding value, so it is a function. Since equation involves square root, quantity under radical sign can't be negative. Thus, 2x - 1 ≥ 0, solve for x. The domain is (½, ∞)

identifying functions from their equations (d)

y ≤ x - 1 Does not define a function. Any number can be used for x The domain is (-∞, ∞)

identifying functions from their equations (c)

y² = x Since one value of x corresponds to two values of y, this is not a function. Because x is equal to the square of y, the values of x must ALWAYS be nonnegative. The domain is [0, ∞)

the range of a constant function f(x) = b

{b}

in a function

⁻no two ordered pairs can have the same first component and different second components ⁻there is exactly one value of the dependent variable (the second component) for each value of the independent variable (the first component)

Variations of the definition of a function

1. A function is a relation in which, for each value of the first component of ordered pairs, there is exactly one value of the second component. 2. A function is a set of ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns exactly one range value to each distinct domain value.

independent variable

(x, y) --- x is the independent variable

dependent variable

(x, y) --- y is the dependent variable

graphs that do not represent functions are still relations

Remember that all equations and graphs represent relations and that all relations have a domain and range.

for every relation

there are two important sets of elements called the domain and range

relations and functions can be described by

using a rule that tells how to determine the dependent variable for a specific value of the independent variable