Understanding Functions: Definition, Domain, Range, and Graphs

1. Section 4-1 • Objectives: • To identify a function • To determine the domain, range, and zeros of a function • To graph a function Functions

2. Introduction • In the past we have studied two specific types of functions (linear and polynomial functions) • We will now examine functions in general • Definition: A function is a correspondence or rule that assigns every element in a set D to exactly one element in a set R. f The function f maps (pairs) each domain element to a unique range element f(x) x f(x) Range element f(x) Domain element x Read “f of x” D (domain) R (range)

3. Introduction • Ex) the function f(x) = 6x • x is a member of the domain • f(x) is the range • f(2) = 6(2) = 12 • 2 is a member of the domain and it is mapped to 12, which is a member of the range. • We can treat a function f as a set of ordered pairs such that x is an element of the domain of f and y is the corresponding element of the range. f = 6x domain range 12 2 x 6x (x,y)

4. Domain and Range • The domain of a function is the set of inputs (x values) for which the function produces a real output. • Consider the functions f(x) = x2 and g(x) = 1/x. • In f(x), any value of x can produce a valid result • since any number can be squared to give a real number • So the domain of f(x) is all values of x (all real numbers) • In g(x), though, not every value of x can generate an output: • When x = 0, g(x) is undefined (1/0). • So the domain of g(x) is x < 0 and x > 0 (x ≠ 0). f(x) = x2 g(x) = 1/x D: All real #’s D: x < 0 and x > 0 (x ≠ 0)

5. Examples: • Give the domain of each function: Recall: 1 - r2 ≥ 0 (1 - r)(1 + r) ≥ 0 Recall: x-7 ≠ 0 To be true, either both have to be positive or both have to be negative +7 +7 Domain: x ≠ 7 Case 1: 1 - r ≥ 0 and 1 + r ≥ 0 r ≤ 1 and r ≥ -1 (All real #’s except 7) Case 2: 1 - r ≤ 0 and 1 + r ≤ 0 r ≥ 1 and r ≤ -1 Your Turn: Problem: r can’t be positive and negative at the same time So domain is: 1 ≥ r ≥ -1

6. Domain and Range • Consider the function • Recall: the square root of a negative number is imaginary • h(x) has a domain of x ≥ 0.

7. Domain and Range - Graphs • We can determine the domain and range of a function if we are given a graph. • Range: • Recall - Range is the set of y coordinates on the graph • we can find the range of y = f(x) by projecting the graph of f onto the y-axis. • Domain: • Recall - Domain is the set of x coordinates on the graph • we can find the domain of y = f(x) by projecting the graph of f onto the x-axis. Range of f Domain of f

8. Domain and Range - Graphs • Examples – use the graph to find the domain, range, and zeros of each function. b) a) 4 2 2 10 2 1 4 7 -1 -1 Domain: Range: Zero(s): All Real Numbers Domain: Range: Zero(s): x ≥ 1 2 ≥ y > -1 y ≤ 4 x = 2 x = 4, x = 7, x = 10

9. More on Domain and Range • Recall: A function is a correspondence or rule that assigns every element in a set D to exactly one element in a set R. • Let f be a function • Let x represent any input value (domain) • Let f(x) represent any output value (range) • The value of f(x) depends on the value of x • So, x is the independent variable and f(x) is the dependent variable • So the range depends on the domain

10. Relations and Functions • Definition: A relation is any correspondence or rule that pairs the members of two sets (the domain and range). • Ex) Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. • Definition: A function is a specific type of relation, such that for each member of the domain is paired with a unique member of the range. • In other words, for each x there can only be one y • Ex) Number of Inches = 12*Number of Feet • For each length in feet there is a unique length in inches

11. Functions • Recall: The vertical line test can be used to determine whether a relation is a function. Examples: Use the Vertical-Line Test to determine whether the following are functions

12. Homework Page 123-124: 1-6 (all), 9, 10, 12, 13-21 (odd), 25 (show work)