1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78 Chapter 0 Functions

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78  Functions and Their Graphs  Some Important Functions  The Algebra of Functions  Zeros of Functions – The Quadratic Formula and Factoring  Exponents and Power Functions  Functions and Graphs in Applications Chapter Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78 § 0.1 Functions and Their Graphs

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78  Rational and Irrational Numbers  The Number Line  Open and Closed Intervals  Applications of Functions  Domain of a Function  Graphs of Functions  The Vertical Line Test  Graphing Calculators  Graphs of Equations Section Outline

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78 DefinitionExample Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern Rational & Irrational Numbers

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78 The Number Line A geometric representation of the real numbers is shown below. The Number Line -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78 Open & Closed Intervals DefinitionExample Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves [-1, 4] -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78 Functions in ApplicationEXAMPLE SOLUTION (Response to a Muscle) When a solution of acetylcholine is introduced into the heart muscle of a frog, it diminishes the force with which the muscle contracts. The data from experiments of the biologist A. J. Clark are closely approximated by a function of the form where x is the concentration of acetylcholine (in appropriate units), b is a positive constant that depends on the particular frog, and R(x) is the response of the muscle to the acetylcholine, expressed as a percentage of the maximum possible effect of the drug. (a) Suppose that b = 20. Find the response of the muscle when x = 60. (b) Determine the value of b if R(50) = 60 – that is, if a concentration of x = 50 units produces a 60% response. This is the given function. (a)

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78 Functions in Application Replace b with 20 and x with 60. CONTINUED Simplify the numerator and denominator. Divide. Therefore, when b = 20 and x = 60, R (x) = 75%. This is the given function. (b) Replace x with 50. Replace R(50) with 60.

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78 Functions in ApplicationCONTINUED Simplify the numerator. Multiply both sides by b + 50 and cancel. Distribute on the left side. Subtract 3000 from both sides. Divide both sides by 60. Therefore, when R (50) = 60, b = 33.3.

11. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78 FunctionsEXAMPLE SOLUTION If, find f (a - 2). This is the given function. Replace each occurrence of x with a – 2. Evaluate (a – 2) 2 = a 2 – 4a + 4. Remove parentheses and distribute. Combine like terms.

12. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 78 Domain DefinitionExample Domain of a Function: The set of acceptable values for the variable x. The domain of the function is

13. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 78 Graphs of Functions DefinitionExample Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xy- plane.

14. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 78 The Vertical Line Test DefinitionExample Vertical Line Test: A curve in the xy-plane is the graph of a function if and only if each vertical line cuts or touches the curve at no more than one point. Although the red line intersects the graph no more than once (not at all in this case), there does exist a line (the yellow line) that intersects the graph more than once. Therefore, this is not the graph of a function.

15. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 78 Graphing Calculators Graphing Using a Graphing Calculator StepDisplay 1) Enter the expression for the function. 2) Enter the specifications for the viewing window. 3) Display the graph.

16. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 16 of 78 Graphs of EquationsEXAMPLE SOLUTION Is the point (3, 12) on the graph of the function ? This is the given function. Replace x with 3. Replace f (3) with 12. Simplify. Multiply. false Since replacing x with 3 and f (x) with 12 did not yield a true statement in the original function, we conclude that the point (3, 12) is not on the graph of the function.