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2.3 Introduction to Functions Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation, and the set of all second components is called the range of the relation. Definition of a Function A function is a correspondence between two sets X and Y that assings to each element x of set X exactly one element y of Y . Domain and Range For each element x in X , the corresponding element y in Y is called the value of the function at x. The set X is called the domain of the function, and the set of all function values, Y , is called the range of the function. Ex 1: Determine whether each relation is a function. a. {(4,5), (6, 7), (8,8)} b. {(5, 6), (4, 7), (6, 6), (6, 7)} Solution We begin by making a figure for each relation that shows set X , the domain, and set Y , the range. Solution for part (a) X Y 4 6 8 5 7 8 Domain Range The figure shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relation have the same first component different second components. Thus, the relation is a function Solution for part (b) X Y 4 5 6 6 7 The figure shows that 6 corresponds to both 6 and 7. Domain Range If any element in the domain corresponds to more than one element in the range, the relation is not a function, Thus, the relation is not a function. Practice Exercises Determine whether each relation is a function. Give the domain and range for each relation. 1. {(7, 7), (5, 5), ( 3, 3), (0, 0)} 2. {(4,1), (5,1), (4, 2)} Answers 1. Domain {7, 5, 3, 0} Range {7, 5, 3, 0} Given relation is a function. 2. Domain {4,5} Range {1, 2} Given relation is not a function. Function Notation y f ( x) The variable x is called the independent variable, because it can be assigned any of the permissible numbers from the domain. The variable y is called the dependent variable, because its value depends on x. Function Notation The special notation f ( x), read "f of x" or "f at x," represents the value of the function at the number x. The notation f ( x) does not mean "f times x." Ex 2: Determine whether each equation defines y as a function of x. 1. x y 25 2. x y 25 2 2 Solution Solve each equation for y in terms of x. If two or more values of y can be obtained for a given x, the equation is not a function. 1. x y 25 y 25 x Solution continued From this last equation we can see that for each value of x, there is one and only one value of y. Thus, the equation defines y as a function of x. Solution of 2 x y 25 2 2 y 25 x 2 2 y 25 x 2 The in this last equation shows that for certain values of x (all values between 5 and 5), there are two values of y. For this reason the equation does not define y as a function of x. Practice Exercises Determine whether each equation difines y as a function of x. 1. x y 25 2 2. 4 x y 2 Answers 1. The equation defines a function. 2. The equation does not define a function. Ex 3: Evaluating a Function If f ( x) x 10 x 3, evaluate: a. f (1) b. f ( x 2) c. f ( x) Solution We substitute 1, x 2, and x for x in the definition of f . 2 Solution part a. We find f (1) by substituting 1 for x in the equation f ( x) x 10 x 3. 2 f ( 1) ( 1 ) 10( 1 ) 3 2 1 10 3 8 Thus f (1) 8. Solution part b. We find f ( x 2) by substituting x 2 for x in the equation f ( x) x 10 x 3. 2 f (x 2) (x 2) 10(x 2) 3 2 f ( x) x 4 x 4 10 x 20 3 2 f ( x) x 6 x 19 2 Solution part c. We find f ( x) by substituting x for x in the equation f ( x) x 10 x 3. 2 f 2 x x ( )( ) 10( x ) 3 f ( x) x 10 x 3 2 Practice Exercise Evaluate the function h( x ) x x 1 at the given values of the independent variable and simplify. a. h(2) b. h(1) c. h( x) d. h(3a) 3 Answer a. 7 b. 1 3 c. x x 1 d. 27a 3a 1 3 Piecewise Functions A function defined by two (or more) equations over a specified domain is called a piecewise function. Ex 4: Evaluating a Piecewise Function Evaluate the piecewise function at the given values of the independent variable. 6 x 1 if x 0 f ( x) 7 x 3 if x 0 a. f (3) b. f (0) c. f (4) Solution part a To find f (3), we let x 3. Because 3 is less than 0, we use the first line of the piecewise function. f ( x) 6 x 1 This is the function's equation for x 0. f (3) 6(3) 1 19 Solution part b To find f (0), we let x 0. Because 0 is equal to 0, we use the second line of the piecewise function. f ( x) 7 x 3 This is the function's equation for x 0. f (0) 7(0) 3 3 Solution part c To find f (4), we let x 4. Because 4 is greater than 0, we use the second line of the piecewise function. f ( x) 7 x 3 This is the function's equation for x 0. f (4) 7(4) 3 31 Practice Exercise Evaluate the piecewise function at the given values of the independent variable. if x 5 x 5 g ( x) ( x 5) if x 5 a. g (0) b. g ( 6) c. g ( 5) Answer a. g (0) 5 b. g (6) 1 c. g (5) 0 Finding a Function’s Domain If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f ( x) is a real number. Finding a Function’s Domain Exclude from a function's domain real numbers that cause division by zero and real numbers that result in an even root of a negative number. Ex 5: Find the domain of each function a. f ( x) 8 x 5 x 2 2 2 b. g ( x) x5 c. h( x) x 2 Solution part a The function f ( x) 8 x 5 x 2 contains 2 neither division nor an even root. The domain of f is the set of all real numbers. Solution part b 2 The function g ( x) contains division. x5 Because division by zero is undefined, we must exclude from the domain values of x that cause x 5 to be 0. Thus x cannot equal to 5. The domain of function g is {x | x 5}. Solution part c The function h( x) x 2 contains an even root. Because only non-negative numbers have real square roots, the quantity under the radical sign, x 2 must be greater than or equal to 0. Thus, x 2 0 or x 2 Therefore the domain of h is {x | x 2} or the interval [2, ). Practice Exercises Find the domain of each function: 12 x 1. h( x) 2 x 36 1 2. f ( x) x2 Answers 1. {x | x 6, x 6} 2. {x | x 2} or ( 2, )