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Pre-Calculus

Pre-Calculus . Chapter 1 Functions and Their Graphs. Warm Up 1.5. Given f ( x ) = 2 x – 3 and g ( x ) = x 2 – 1 , find: f ( x ) + g ( x ) = f ( x ) – g ( x ) = f ( x ) · g ( x ) = f ( x ) / g ( x ) =. 1.5 Combinations of Functions. Objectives :

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Pre-Calculus

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  1. Pre-Calculus Chapter 1 Functions and Their Graphs

  2. Warm Up 1.5 • Given f (x) = 2x – 3 and g(x) = x2 – 1, find: • f (x) + g(x) = • f (x) – g(x) = • f (x) · g(x) = • f (x) / g(x) =

  3. 1.5 Combinations of Functions • Objectives: • Add, subtract, multiply, and divide functions. • Find compositions of one function with another function. • Use combinations of functions to model and solve real-life problems.

  4. Vocabulary • Arithmetic Combination of Functions • Composition of Functions

  5. Arithmetic Operations • Let f and g be two functions with overlappingdomains. Then, for all x common to both domains: • (f + g)(x) = f (x) + g(x) • (f – g)(x) = f (x) – g(x) • (f g)(x) = f (x) · g(x)

  6. Example 1 • Let f (x) = 2x +1 and g (x) = x2 + 2x – 1. Find (f – g)(x) algebraically and graphically. Then find (f – g)(2).

  7. Example 2 • Find (f /g)(x) and (g/f )(x) for the functions given by and . • Specify the domain for each quotient.

  8. Consider this … • If you drop a pebble into a pond, a circular ripple extends out from the drop point. The radius of the circle is a function of time. The area enclosed by the circular ripple is a function of the radius.

  9. What If …? • Suppose that the radius is increasing at a constant rate of 8 in/sec. Then r = 8t • What is the radius at t = 5? r = __________________ • The formula for the area of the circular region is A = ____________________ • What is the area at t = 5? A = ____________________

  10. How Are the Variables Related? • Time is the input for the radius function. • And the radius is the input for the area function. Time Radius Area A(r) or A(r(t)) r(t)

  11. This means … • Area is a function of time through this chain: • Area depends on radius. • Radius depends on time. • Area is a composite function of time, A(r(t)).

  12. Composition of Functions • The composition of the function fwith the function gis

  13. Example 3 • Find the following values of the using the values of f and g found in the table. • Find f (g(2). • Find g(f (2).

  14. Example 4 • Let and . Find . Then find, if possible, and .

  15. Homework 1.5.1 • Worksheet 1.5 #1, 7, 9, 17, 23, 35, 37, 51, 53, 73, 74

  16. r Warm Up 1.5 • A square concrete foundation was prepared as a base for a large cylindrical gasoline tank as shown. • Write the radius r of the tank as a function of the length x of the sides of the square. • Write the area A of the circular base of the tank as a function of the radius r.

  17. In-Class Assignment • Work with a partner to complete the worksheet “Domain of Composite Functions”.

  18. Domain of Composite Functions • The domain of f (g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. • The outputs of g must be restricted so that they are in the domain of f.

  19. Finding the Domain • Let and . Find the domain of . The outputs of g can be any real number. But the domain of f is restricted to all real numbers ≠0. Therefore, or . So, domain of is all real numbers except .

  20. Example 4 • Find the domain of the composition for the functions given by and .

  21. Example 5 • Does ? • Let and . • Let and .

  22. Special Case • If (f ○g)(x) = (g ○ f)(x), • then (f ○g)(x) = x • and(g ○ f)(x) = x.

  23. Application Problem • The number N of bacteria in a refrigerated food is given by N(T) = 20T2 – 80T + 500, 2 ≤ T ≤ 14 where T is the temperature of the food in ˚C. When the food is removed from the refrigerator, the temperature of the food is given by T(t) = 4t + 2, 0≤ t ≤ 3 where t is the time in hours. • Find the composition N(T(t)) & interpret its meaning. • Find the number of bacteria in the food when t = 2 hours. • Find the time when the bacterial count reaches 200.

  24. Homework 1.5.2 • Worksheet 1.5 # 39 – 49 odd, 63 – 71 odd, 77

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