SECTION 2.1 Functions 73 The domain of f g consists of all real numbers x for which ( ) ≠ g x 0 that are also in the domains of both f and g. Since ( ) = g x 0 when = x 0, exclude 0 as well as −2 and 1 from the domain. The domain of f g is { } ≠ − ≠ ≠ x x x x 2, 0, 1 . SUMMARY Function • A relation between two sets of real numbers so that each number x in the first set, the domain, corresponds to exactly one number y in the second set. • A set of ordered pairs ( ) x y , or ( ) ( ) x f x , in which no first element is paired with two different second elements. • The range is the set of y -values of the function that are the images of the x -values in the domain. • A function f may be defined implicitly by an equation involving x and y or explicitly by writing ( ) = y f x . Unspecified domain If a function f is defined by an equation and no domain is specified, then the domain is the largest set of real numbers for which ( ) f x is a real number. Function notation • ( ) = y f x • f is a symbol for the function. • x is the independent variable, or argument. • y is the dependent variable. • ( ) f x is the value of the function at x. Now Work PROBLEM 71 In calculus, it is sometimes helpful to view a complicated function as the sum, difference, product, or quotient of simpler functions. For example, ( ) = + F x x x 2 is the sum of ( ) = f x x2 and ( ) = g x x. ( ) = − + H x x x 1 1 2 2 is the quotient of ( ) = − f x x 1 2 and ( ) = + g x x 1. 2 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 1. The inequality − < < x 1 3 can be written in interval notation as . (pp. A72–A74) 2. If = − x 2, the value of the expression − + x x x 3 5 1 2 is . (pp. A6–A7) 3. The domain of the variable in the expression − + x x 3 4 is . (p. A7) 4. Solve the inequality: − > x 3 2 5. Graph the solution set. (pp. A76–A77) 5. To rationalize the denominator of − 3 5 2 , multiply the numerator and denominator by . (p. A89) 6. A quotient is considered rationalized if its denominator has no . (p. A89) 2.1 Assess Your Understanding 7. For a function ( ) = y f x , the variable x is the variable, and the variable y is the variable. 8. Multiple Choice The set of all images of the elements in the domain of a function is called the . (a) range (b) domain (c) solution set (d) function 9. Multiple Choice The independent variable is sometimes referred to as the of the function. (a) range (b) value (c) argument (d) definition 10. True or False The domain of f g consists of the numbers x that are in the domains of both f and g. 11. True or False Every relation is a function. 12. Four ways of expressing a relation are , , , and . 13. True or False If no domain is specified for a function f, then the domain of f is the set of real numbers. 14. True or False If x is in the domain of a function f, we say that f is not defined at x, or ( ) f x does not exist. Concepts and Vocabulary 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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