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2.2, 2.3 Functions

2.2, 2.3 Functions. Function is a corresponding between 2 sets: domain and range, such that each member of the domain correspond to exactly one member of the range For example: Each person corresponds to his or her biological mother Each person corresponds to his or her weight

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2.2, 2.3 Functions

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  1. 2.2, 2.3 Functions

  2. Function is a corresponding between 2 sets: domain and range, such that each member of the domain correspond to exactly one member of the range • For example: • Each person corresponds to his or her biological mother • Each person corresponds to his or her weight • Each natural number (1, 2, 3, 4…) corresponds to the square of that number (1, 4, 9, 16…) Domain correspondence Range

  3. In a set of ordered pairs, the domain is the set of all first coordinate (x), and the range is the set of all second coordinate (y) • For example: {(1,-1), (2,-2),(3,-3),(4,-4)} • The domain is {1,2,3,4} • The range is {-1,-2,-3,-4}

  4. Determine whether each of the following is a function. If yes, list the domain and the range 1) {(Sue, 18 years old), (Peter,19 years old),(Kim, 16 years old), (Sue, 20 years old)} • This is not a function because Sue corresponds to two numbers: 18 and 20 years olds 2) {(1,3), (2,3), (3,4)} • This is a function. The domain is {1,2,3}, and the range is {3,4} 3) y = x3 • This is a function. The domain is {1,2,3,4,…} and the range is {1,8,27,64…}

  5. NO, because z corresponds to both X and Z YES, each element in the domain corresponds to only one element in the range

  6. Determine a function by The Vertical-Line Test: if the vertical line cross the graph more than once, then the graph is not a function no yes yes

  7. Function notation: f(x) read f of x Ex: f(x) = 2x Imagine this function is a change machine. If we put $1 bill in the machine, it will give out 2 coins of 50cents. x (number of dollar bills) f(x) number of coins INPUT OUTPUT 2 4 3 6 4 8 Find f(6), f(a + 1) if f(x) = 2x F(6) = 2 * 6 =12 F(a + 1) = 2 * (a + 1) = 2a + 2

  8. Ex2: F(n) = 3n2 – 2n Find f(0), f(-1), f(2a), 3f(a) • F(0) = 3*02 – 2*0 = 0 - 0 = 0 • F(-1)= 3(-1 )2 – 2(-1) = 3 + 2 = 5 • F(2a) = 3(2a)2 – 2(2a) = 3 * 4a2 - 4a = 12a2 – 4a • 3 f(a) = ? F(a) = 3*a2 – 2*a = 3a2 – 2a 3f(a) = 3 (3a2 – 2a ) = 9a2 – 6a

  9. Find the domain and the range for each function Domain: all real numbers (-∞, ∞) Domain: all real numbers (-∞, ∞) Range: [-4, ∞) Range: all real numbers (-∞, ∞)

  10. Domain [-3, 3) Domain: (-5, 4) Range: (-5, 5) Range: [-2, 2)

  11. More problems with domain • f(x) = x + 1 Domain is (-∞,∞) interval notation 2) f(x) = 2x2 + 1 x + 2 Domain is all real numbers except -2 (-∞,-2) U (-2, ∞) interval notation

  12. Credit card debt in the US from 1992 through 1999 is modeled by this equation: y = 47.3x + 281 (in billion) where x = 0 represents for 1992 • Approximate the credit card debt in 1992, 1993, and 1999 using the equation • Graph the linear equation using the information from a • Use the graph to approximate the credit card debt in 1996

  13. y = 47.3x + 281 (in billion dollars) For 1992, x = 0 So y = 47.3(0) + 281 = 281 For 1993, x = 1 So y = 47.3(1) + 281 = 328.3 For 1999, x = 7 So y = 47.3(7) + 281 = 612.1 For 1996, look at the graph, we have y = 470 billion dollars

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