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Chapter 0 Functions

Chapter 0 Functions. Chapter Outline. 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. § 0.1. Functions and Their Graphs.

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Chapter 0 Functions

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  1. Chapter 0Functions

  2. Chapter Outline • 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

  3. §0.1 Functions and Their Graphs

  4. Section Outline • 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

  5. Rational & Irrational Numbers

  6. The Number Line -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  7. Open & Closed Intervals -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  8. Functions in Application EXAMPLE (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. SOLUTION (a) This is the given function.

  9. Functions in Application CONTINUED Replace b with 20 and x with 60. Simplify the numerator and denominator. Divide. Therefore, when b = 20 and x = 60, R(x) = 75%. (b) This is the given function. Replace x with 50. Replace R(50) with 60.

  10. Functions in Application CONTINUED 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. Functions EXAMPLE If , find f (a - 2). SOLUTION This is the given function. Replace each occurrence of x with a – 2. Evaluate (a – 2)2 = a2 – 4a + 4. Remove parentheses and distribute. Combine like terms.

  12. Domain

  13. Graphs of Functions

  14. The Vertical Line Test

  15. Graphing Calculators

  16. Graphs of Equations EXAMPLE Is the point (3, 12) on the graph of the function ? SOLUTION This is the given function. Replace x with 3. Replace f (3) with 12. Simplify. false Multiply. 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.

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