Chapter 1 Functions and Their Graphs

1.2.1 Introduction to Functions Objectives:  Determine whether relations between two variables represent a function.  Use function notation and evaluate functions.  Find the domains of functions. 2

Warm Up Solve Algebraically and Graphically A runner runs at a constant rate of 4.9 miles per hour. a. Determine how far the runner can run in 3.1 hours. b. Determine how long it will take to run a 26.2-mile marathon. VVerbal model: Distance = Rate * Time AAlgebraic equation: d = 4.9t 3

Vocabulary Relation Function Domain Range Independent Variable Dependent Variable Function Notation Implied Domain 4

Representations of Functions If you pour a cup of coffee, it cools more rapidly at first, then less rapidly, finally approaching room temperature. Since there is one and only one temperature at any one given time, we can say that temperature is a _____________ of time. 5

Representation of Functions Functions can be represented:  Graphically  Algebraically  Numerically  Verbally 6

Graphically - Temperature y (°C) as a function of x (minutes). 7 Room Temp.

Algebraically Algebraic Equation y = (0.8) x 8

Numerically Use equation or TABLE feature of graphing calculator. x (min.) y (°C)

Verbally If you pour a cup of coffee, it cools more rapidly at first, then less rapidly, finally approaching room temperature. 10

Variables In our coffee example, which is the dependent variable and which is the independent variable? Why? The temperature depends on the amount of time the coffee has been cooling. Temperature  Dependent Time  Independent 11

Domain and Range Domain The set of values of the independent variable. (all “legal” values of x ) Range The set of values of the dependent variable. (all “legal” values of y )  What are the domain and range in our example? 12

Example 1 Function or not? a. x y

Example 2 Function or not? a. x 2 + y = 1 b. –x + y 2 = 1 14

Function Notation A function is denoted by the symbol f (x), “ f of x ” or “ the value of f at x ”. So, y = f (x). 15

Example 3 Solve for each if g(x) = –x 2 + 4x g(2) = 2. g(t) = 3. g(x + 2) = 16

Domain of a Function We can specify the domain by what it is or by what it is not. Explicit Domain Ex. The set of all real numbers. Implicit Domain Ex. x ≠ 0. This implies all real numbers except x = 0. 17

Example 4 Let.  What values of x make this function undefined? Why?  What is the domain of this function? 18

Example 5 Let.  What values of x make this function undefined? Why?  What is the domain of this function? 19

Domain in General The domain of many functions is the set of all real numbers. However, we cannot:  Divide by zero  Have a negative number in a square root (or other even root). 20

Domain Notation The set of all real numbers. –∞ < x < ∞ or (–∞, ∞) Exclude a value of x. x ≠ a or (–∞, a) U (a, ∞) An interval of x. a ≤ x < b or [a, b) 21

Interval Notation [a, b]  a ≤ x ≤ b. [a, b)  a ≤ x < b. (a, b]  a < x ≤ b. (a, b)  a < x < b. 22

Example 6 23

Homework Worksheet  # 1 – 7 odd, 13, 17, 19, 27 – 33 odd 24

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