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1.5 Combinations of Functions

Learn how to find sum, difference, product, and quotient of functions, evaluate compositions, and decompose functions in this comprehensive guide. Practice exercises included.

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1.5 Combinations of Functions

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  1. 1.5 Combinations of Functions

  2. Sum, Difference, Product, and Quotient of Functions • Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows. • Sum: (f + g)(x) = f(x) + g(x) • Difference: (f – g)(x) = f(x) – g(x) • Product: (fg)(x)= f(x)*g(x) • Quotient: (f/g)(x) = f(x)/g(x)

  3. Finding the Sum and Difference of Two Functions • Given f(x) = 2x + 1 and g(x) = x2 + 2x -1, find (f + g)(x) and (f – g)(x). Then evaluate the sum and difference when x = 2.

  4. Finding the Product of Two Functions • Given f(x) = x2 and g(x) = x – 3. find (fg)(x). Then evaluate the product when x = 4.

  5. Finding the Quotient of Two Functions • Find (f/g)(x) and (g/f)(x) for the functions given by and Then find the domains of f/g and g/f.

  6. Composition of Functions • Value fed to first function • Resulting value fed to second function  • End result taken from second function 

  7. Definition • Notation for composition of functions: • Alternate notation:

  8. Composition of Functions • Given f(x) = x + 2 and g(x) = 4 – x2, evaluate • f(g(x)) • g(f(x)) When x = 0, 1, 2, and 3.

  9. A Case in Which f(g(x)) = g(f(x)) • Given f(x) = 2x +3 and g(x) = .5(x – 3) • f(g(x)) • g(f(x))

  10. Decomposition of Functions Someone once dug up Beethoven's tomb and found him at a table busily erasing stacks of papers with music writing on them.  They asked him ... "What are you doing down here in your grave?"  He responded, "I'm de-composing!!" But, seriously folks ... Consider the following function which could be a composition of two different functions. Look for the “inner” and “outer” function.

  11. Decomposition of Functions • The function could be decomposed into two functions, k and j

  12. Bacteria Count The number N of bacteria in a refrigerated food is given by where T is the temperature of the food (in degree Celsius). When the food is removed from refrigeration, the temperature of the food is given by where t is the time (in hours). • Find N(T(t)) and interpret its meaning in context. • Find the number of bacteria in the food when t=2 hours • Find the time when the bacterial count reaches 2000.

  13. Homework • Page 58-61 5-11 odd, 14-26 even, 35-38 all, 56 part a and b, 66-70 even, 81

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