Section 4 1 Functions Objectives To identify a
Section 4 -1 Functions Objectives: • To identify a function • To determine the domain, range, and zeros of a function • To graph a function
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 x Domain element x D (domain) The function f maps (pairs) each domain element to a unique range element f(x) R (range) Range element f(x) Read “f of x”
Introduction • Ex) the function f(x) = 6 x • 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 f = 6 x range. domain range (x, y) x 2 6 x 12
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) = x 2 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) = x 2 D: All real #’s g(x) = 1/x D: x < 0 and x > 0 (x ≠ 0)
Examples: • Give the domain of each function: Recall: 1 - r 2 ≥ 0 Recall: x-7 ≠ 0 +7 +7 Domain: x≠ 7 (All real #’s except 7) Your Turn: (1 - r)(1 + r) ≥ 0 To be true, either both have to be positive or both have to be negative Case 1: 1 - r ≥ 0 and 1 + r ≥ 0 r ≤ 1 and r ≥ -1 Case 2: 1 - r ≤ 0 and 1 + r ≤ 0 r ≥ 1 and r ≤ -1 Problem: r can’t be positive and negative at the same time So domain is: 1 ≥ r ≥ -1
Domain and Range • Consider the function – Recall: the square root of a negative number is imaginary – h(x) has a domain of x ≥ 0.
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
Domain and Range - Graphs • Examples – use the graph to find the domain, range, and zeros of each function. b) a) 4 2 2 2 -1 Domain: Range: Zero(s): All Real Numbers 2 ≥ y > -1 x=2 1 4 Domain: x≥ 1 -1 Range: Zero(s): 7 10 y≤ 4 x = 4, x = 7, x = 10
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
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
Functions • Recall: The vertical line test can be used to determine whether a relation is a function. The Vertical-Line Test If no vertical line intersects a given graph in more than one point, then the graph is the graph of a function Examples: Use the Vertical-Line Test to determine whether the following are functions
Homework Page 123 -124: 1 -6 (all), 9, 10, 12, 13 -21 (odd), 25 (show work)
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