Functions+Notes+_updated_.pdf

B what is the domain of f solution a taking a look to

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b. What is the domain of f ? Solution: a. Taking a look to the graph of f we can conclude that Dom( f ) = ( -∞ , 0) [1 , + ) . b. Since the graph f has four different sections, we will consider four cases: F The first piece corresponds to the real numbers x < - 3. In this case, the graph is a line y = ax + b . Notice that ( - 5 , - 1) and ( - 4 , 0) are two points of such a line. Thus: - 5 a + b = - 1 - 4 a + b = 0 Solving the system above we get that a = 1, b = 4. F The second piece in the graph of f is just the point ( - 3 , - 2). F The next piece ( - 3 < x < 0) is just the constant line y = 1.
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2. Functions 11 F The last piece ( x 1) is a parabola y = ax 2 + bx + c with vertex (2 , - 4). Since the vertex’s x -coordinate is - b 2 a , we have that - b 2 a = 2 On the other hand, since (2 , - 4) and (4 , 0) are in the parabola, we have that b = - 4 a 4 a + 2 b + c = - 4 16 a + 4 b + c = 0 The solution is a = 1, b = - 4, c = 0. Therefore, the technology formula of f is as follows: f ( x ) = x + 4 if x < - 3 - 2 if x = - 3 1 if - 3 < x < 0 x 2 - 4 x if x 1 Example 2.6. Let f be the function specified by f ( x ) = x 2 if - 2 < x 0 1 x if 0 < x 4 a. What is the domain of f ? b. Find f (0) and f (1) . c. Sketch the graph of f . Solution: a. The domain of f is ( - 2 , 4], because f ( x ) is specified only when - 2 < x 4. b. f (0) = 0 2 = 0 and f (1) = 1 1 = 1.
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12 J. S´ anchez-Ortega c. -2 -1 1 2 3 4 0 1 2 3 4 x y 2.4 Linear Functions Linear functions are among the simplest functions. A function f is said to be a linear function if f ( x ) = mx + b , where m and b are fixed real numbers. For example, f ( x ) = 3 x - 1 , y = 3 x - 1 . The Change in a Quantity: Delta Notation If a quantity q changes from q 1 to q 2 , the change in q is just the difference: Change in q = Second value - First value = q 2 - q 1 . In Mathematics, we use the letter Δ to stand for change, and we write the change in q as Δ q . Δ q = Change in q = q 2 - q 1 . Examples 2.7. Let y = 3 x - 1 . If x is changed from 1 to 3, we write Δ x = Second value - First value = 3 - 1 = 2. Looking at our linear function, we see that when x changes from 1 to 3, y changes from 2 to 8. So, Δ y = Second value - First value = 8 - 2 = 6.
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2. Functions 13 Using delta notation, we can now write, for our linear function, Δ y = 3Δ x, or Δ y Δ x = 3 . Change in y = 3 × Change in x Because the value of y increases by exactly 3 units for every increase of 1 unit in x , the graph is a straight line rising by 3 units for every 1 unit we go to the right. We then say that we have a rise of 3 units for each run of 1 unit. Notice that m = 3 is a measure of the steepness of the line ; we call m the slope of the line: Slope = m = Δ y Δ x = Rise Run The Roles of m and b in the linear function f ( x ) = mx + b Role of m Numerically. If y = mx + b then y changes m units for every 1- unit changes in x . A change of Δ x unites in x produces a change of Δ y = m Δ x units in y . Thus, m = Δ y Δ x = Change in y Change in x Graphically. m is the slope of the line y = mx + b : m = Δ y Δ x = Rise Run = Slope For positive m , the graph rises m units for every 1-unit move to the right; for negative m , the graph drops | m | units for every 1-unit move to the right.
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14 J. S´ anchez-Ortega Role of b Numerically. When x = 0, y = b .
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