958 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function SUMMARY Library of Functions: Continuity Properties Function Domain Property • Polynomial function All real numbers Continuous at every number in the domain • Rational function R x P x Q x , ( ) ( ) ( ) = P , Q are polynomials x Q x 0 { ( ) } ≠ Continuous at every number in the domain; hole or vertical asymptote where R is undefined • Exponential function All real numbers Continuous at every number in the domain • Logarithmic function Positive real numbers Continuous at every number in the domain • Sine and cosine functions All real numbers Continuous at every number in the domain • Tangent and secant functions All real numbers, except odd integer multiples of 2 π Continuous at every number in the domain; vertical asymptotes at odd integer multiples of 2 π • Cotangent and cosecant functions All real numbers, except integer multiples of π Continuous at every number in the domain; vertical asymptotes at integer multiples of π ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 14.3 Assess Your Understanding 1. For the function f x x x x x x x if 0 1 if 0 2, 5 if 2 5 2 ( ) = ≤ + < < − ≤ ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ find f 0( ) and f 2 . ( ) (pp. 105–107) 2. What are the domain and range of f x x ln ? ( ) = (p. 316 ) 3. True or False The domain of any exponential function ( ) = > ≠ f x a a a , 0; 1, x is all real numbers. (pp. 300–301) 4. Name the trigonometric functions that have asymptotes. (pp. 428, 430, 443–448) 5. True or False Some rational functions have holes in their graph. (pp. 247–254) 6. True or False The functions R x x x 9 3 2 ( ) = − + and p x x 3 ( ) = − are equal. (pp. 236–239) 7. If we approach c from only one side, then we have a(n) limit. 8. The notation is used to describe the fact that as x gets closer to c but remains greater than c , the value of f x( ) gets closer to R . 9. If f x f c lim , x c ( ) ( ) = → then f is at . 10. True or False For any function f, f x f x lim lim x c x c ( ) ( ) = → → − + . 11. True or False If f is continuous at c , then f x f c lim x c ( ) ( ) = → + . 12. True or False Every polynomial function is continuous at every real number. Concepts and Vocabulary In Problems 13–32, use the accompanying graph of y f x . ( ) = 13. What is the domain of f ? 14. What is the range of f ? 15. Find the x -intercept(s), if any, of f. 16. Find the y -intercept(s), if any, of f. 17. Find f 8 ( ) − and f 4 . ( ) − 18. Find f 2( ) and f 6 . ( ) 19. Find f x lim x 6 ( ) →− − 20. Find f x lim . x 6 ( ) →− + 21. Find f x lim . x 4 ( ) →− − 22. Find f x lim . x 4 ( ) →− + 23. Find f x lim . x 2 ( ) → − 24. Find f x lim . x 2 ( ) → + Skill Building y 2 4 2 (2, 3) (6, 2) 4 6 x 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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