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FUNCTIONS Section 3.1

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RELATIONS Definition: A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x → y, then we say that “x” correspond to “y” or that “y” depends on “x”

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Example: Relation X=2 Input Y=5 Output Y= 3X – 1 If X =2 then Y= 3.2 – 1 So, y=5

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A map : illustrate a relation by using a set of inputs and drawing arrows to the corresponding elements in the set of outputs. Ordered pairs can be used to represent x→y as ( x, y ) { (0,-2),(0,1),(1,2),(2,1), (3,4)}

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Determine whether a relation represents a Function Let X and Y be two nonempty sets. “ A function from X into Y is a relation that associates with each element of X exactly one element of Y”

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Let’s now use ordered pairs to identify which of these sets are relations or functions: {(1,4), (2,5), (3,6), (4,7)} Do={1,2,3,4} Rg={4,5,6,7} {(1,4),(2,4)(3,5),(6,10)} Do={1,2,3,6} Rg={4,5,10} { (-3,9), (-2,4), (0,0), (1,1), (-3,8)}

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Determine whether an Equation is a Function Determine if the equation y=2x – 5 defines y as a function of x If x=1, then y=2(1) – 5 = -3 If x=3, then y= 2(3) – 5 = 1 The equation is a FUNCTION

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Example 2 Determine if the equation x 2 +y 2 =1 defines y as a function of x. Solve for y: y 2 = 1 - x 2 y= ± \̸ 1-x 2 If x=0 then y = ±1 This means the equation x 2 +y 2 =1 does not define a function

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Find the value of a Function y = f(x) read “f of x” Example: y=f(x) = 2x – 5 then f(1/2)=2.1/2 – 5 f(1/2)= -4 The variable x is called independent variable or argument, and y is called dependent variable

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Finding the Domain of a Function The domain of a function is the largest set of Real numbers for which the value f(x) is a Real number. Examples: Find the domain of each of the following functions: (a) f(x)= x 2 +5x (b) g(x)= 3x. x 2 -4 (c) h(t) = \̸ 4-3t

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Solutions: (a) Domain of f is the set of all Real Numbers. (b) Domain of g is {x ̸ x ≠±2} (c) Domain of h is { t ̸ t≤4/3}

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Tips to find the Domain of a function Start with the domain as the set of real numbers. If the equation has a denominator, exclude any numbers that give a zero denominator. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative.

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SUMMARY Function: a relation between two sets of real numbers so that each number x in the first set, the domain, has corresponding to it exactly one number y in the second set, the range. Unspecified Domain: If a function f is defined by an equation and no domain is specified, then the domain will be taken to be the largest set of real numbers for which the equation defines a real number. Function Notation: y= f(x) f is the symbol for the variable, x is the independent variable or argument, y is the dependent variable, and f(x) is the value of the function at x, or the image of x.

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GAME TIME DOMAINRANGE

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f(x)={(1,2);(3,4);(-1,0)} ANSWER

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Do = { 1, 3, 4}

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f(x) = 2X + 1 ANSWER

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Do= all real numbers

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g(x) = 1. X - 1 ANSWER

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Do ={ X/ X≠1}

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.. h(x) = √ X-2 ANSWER

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Do={ x/x ≥ 2}

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f(x) = { (1,2); (3,4) ;(-1,0)} ANSWER

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Rg = { 2, 4, 0}

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f(x) = 2X + 1 ANSWER

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Rg = all real numbers

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g(x) = 1. X - 1 ANSWER

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Rg = {x/x≠0}

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.. h(x) = √ X-2 ANSWER

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Rg={x/x≥0}

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