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Understanding Functions

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The set of all the x-values is called the Domain of the function. For each element x in the domain, the corresponding element y is called the image of x. The set of all images of the elements of the domain is called the Range of the function. A function is a rule or a correspondence that associates each x-value with exactly one y-value.

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4 ways to describe a function Mapping Diagram Ordered pairs/Table of values Graph Rule (equation)

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Example: M is the Mother Function Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue Humans Mothers 1. Function as a Mapping.

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M: Mother function Domain of M { Joe, Samantha, Anna, Ian, Chelsea, George } Range of M { Laura, Julie, Hilary, Barbara } In function notation we can write: M( Anna ) = Julie or M( George ) = Barbara Also, if we are told M( x ) = Hilary, That means that x must be = Chelsea

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For the function f below, evaluate f at the indicated values and find the Domain and Range of f 12345671234567 10 11 12 13 14 15 16 f(1) f(2) f(3)f(4) f(5)f(6) f(7) Domain of f: Range of f: {1, 2, 3, 4, 5, 6, 7} {10, 12, 13, 15}

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2. Function as a Set of Ordered Pairs A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components. A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components. In other words, you can’t have two different y-values for the same x-value.

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For each x, there is one related y-value h:{(-2,3), (1,3), (4,5), (10,5)} j:{(1,-2), (2,2), (3,1), (4,-2)} p:{(0,0), (1,1)} What is h(1)? What is j(1)? What is p(1)? For what values is h(x) = 5?

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The mother function M can also be written as ordered pairs M = { (Joe, Laura), (Samantha, Laura), (Anna, Julie), (Ian, Julie), (Chelsea, Hillary), (George, Barbara) }

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3. Function as a Graph Another way to depict a function, is to display the ordered pairs on a graph on the coordinate plane, with the x-values along the horizontal axis, and the y-values on the vertical axis.

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f = {(-3, -1), (-2, -3), (-1, 2), (0, -1), (1, 3), (2, 4), (3, 5)} is graphed below. Domain of f = {-3, -2, -1, 0, 1, 2, 3} Range of f = {-3, -1, 2, 3, 4, 5}

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4. Function Defined by a Rule Let f be a function, consisting of ordered pairs where the second element of the ordered pair is the square of the first element. Some of the ordered pairs in f are (1, 1) (2, 4), (3, 9), (4, 16),……. f is best defined by the rule f(x) = x²

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Function Notation Function Notation f(x) Functions defined on infinite sets are denoted by algebraic rules. Examples of functions defined on all Real numbers f(x) = x² g(x) = 2x – 1 h(x) = x³ The symbol f(x) represents the y-value in the Range corresponding to the Domain value x. The point (x, f(x)) belongs to the function f.

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Evaluating functions Determine the function values (y-values) for the given x-values. 5 -11 -7.5 5 Undefined 2 3 If x is in the denominator, or in a square root, there will be restrictions on the Domain.

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Graph of a function E.g.: The graph of the function f(x) = 2x – 1 is the graph of the equation y = 2x – 1, which is a line. Each point on the line is (x, f(x)) The graph of the function f(x) is the set of points (x, y) in the plane that satisfies the relation y = f(x).

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Domain and Range from a Graph Remember: Domain is the set of all x-values. On a graph, it is represented by all the values from left to right. Range is the set of all the y-values. On a graph, it is represented by all the values from bottom to top. For Real numbers, we write the Domain and Range in interval notation. [ #, # ]

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Domain and Range from a Graph Domain: x [-4, + [ Range: y [-3, + [ 4 0 -4 (-4, 2) x y 4 -4

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The Zero of a Function The zero of a function is the place where the function hits the x-axis. It is the x-intercept. 2 0 -2 x y 2 What is the zero of the function graphed at the right?

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The y-intercept of a Function The y-intercept of a function is the place where the function hits the y-axis. What is the y-intercept of the function graphed at the right? 2 0 -2 x y 2

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Calculating the zero and y-intercept of a function. Calculate the zero of a function by making the function equal to zero and solving for x. Calculate the y-intercept by finding f(0). Given f(x) = 2x + 10, find: a) the zero b) the y-intercept. f(x) = 2x + 10 = 0 2x = -10 x = -5 f(0) = 2(0) + 10 = 10 y = 10

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Calculate the y-intercept of g: Calculate the zeros of g: Consider the function: g(x) = x 2 + 3x – 4 g(0) = (0) 2 + 3(0) – 4 = -4 g(x) = x 2 + 3x – 4 = 0 (x + 4)(x – 1) = 0 x = -4 or x = 1

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Consider the function: g(x) = x 2 + 3x – 4 g(0) = -4 The zeros are x = -4 or x = 1 g

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5 Sign of the function A function is positive where the graph is above the x-axis. It’s negative where the graph is below. x y positive negative -3 The function is positive on the interval x [-3, 5] The function is negative on the intervals x ]- , -3] [5, + [ 1

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Intervals of Increase or Decrease 5 x y -3 1 We need to identify where the function is increasing or decreasing Increasing: x ]- , 1] Decreasing: x [1, + [

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Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph. 4 0 -4 (2, 3) (7, -2.5) x y

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Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph. x 4 0 -4 (2, 3) (7, -2.5) y Domain: Range: y-int: Zeros: Positive: Negative: Increasing: Decreasing: Extrema (max/min):

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Theorem Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

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x y Not a function.

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x y Function.

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4 0 -4 (2, 3) x y Is this a graph of a function?

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