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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”**Example:**Y= 3X – 1 If X =2 then Y= 3.2 – 1 So, y=5**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)}**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”**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)}**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**Example 2**Determine if the equation x2+y2=1 defines y as a function of x. Solve for y: y2= 1 - x2 y= ± \̸ 1-x2 If x=0 then y = ±1 This means the equation x2+y2=1 does not define a function**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**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)= x2+5x (b) g(x)= 3x . x2-4 (c) h(t) = \̸ 4-3t**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}**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.**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.**GAME TIME**10 10 20 20 30 30 40 40**f(x)={(1,2);(3,4);(-1,0)}**ANSWER**f(x) = 2X + 1**ANSWER**g(x) = 1 .**X - 1 ANSWER**. .**• h(x) = √ X-2 ANSWER**f(x) = 2X + 1**ANSWER**g(x) = 1 .**X - 1 ANSWER**. .**• h(x) = √ X-2 ANSWER