Functions+Notes+_updated_.pdf

5 solution the domain of f is r since we can compute

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5) . Solution: The domain of f is R , since we can compute f ( x ) for every real number x . Moreover: f ( - 2) = 3 ( - 2) 2 + 2 = 3 6 = 1 2 , f (0) = 3 0 2 + 2 = 3 2 , f ( 5) = 3 ( 5) 2 + 2 = 3 7 Example 2.3. Are the functions below equal? Support your an- swer! f ( x ) = r x x - 1 , g ( x ) = x x - 1 Solution: The algebraic expressions of f and g are the same, but does it allow us to conclude that f and g are equal? Let us take a look to their domains! Dom f = x R : x x - 1 0 = ( -∞ , 0] (1 , + ) Dom g = { x R : x 0 } ∩ { x R : x > 1 } = (1 , + ) Since their domains do not coincide, we can conclude that f and g are not the same. For example, we could compute f (0) = 0 but notice that g (0) is not defined!
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2. Functions 5 2.2 Vertical-Line Test Given a function f , every point in the graph of f has the form ( x, f ( x )) for some x in the domain of f . Because f assigns a single value f ( x ) to each value of x in the domain , it follows that, in the graph of f , there should be only one y corresponding to any such value of x , namely, y = f ( x ) . In other words, the graph of a function can not contain two or more points with the same x -coordinate , that is, two or more points on the same vertical line . On the other hand, a vertical line at a value of x mot in the domain will contain any points in the graph. This gives us the following rule: for a graph to be the graph of a function, every vertical line must intersect the graph in at most one point.
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6 J. S´ anchez-Ortega In what follows, we list some of the most common types of functions that are often used to model real world situations.
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2. Functions 7
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8 J. S´ anchez-Ortega 2.3 Piecewise-defined functions Piecewise-defined functions are those which are specified by two or more different formulas. For example, the absolute value f ( x ) = | x | is the most common piecewise-defined function: f ( x ) = x if x 0 - x if x < 0 A more complicated piecewise-defined function is the following: Example 2.4. Let f be the function specified by f ( x ) = - 1 if x < 0 x - 1 if 0 < x < 1 x 2 if x 1
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2. Functions 9 a. What is the domain of f ? b. Find f ( - 2) and f (3) . c. Sketch the graph of f . Solution: a. The domain of f is R - { 0 } because f ( x ) is defined for all real numbers x , except for 0. b. To calculate f ( - 2) we need to use the first formula, so f ( - 2) = - 1. To find f (3) we will use the latest formula because 3 > 1; so f (3) = 3 2 = 9. c. To sketch the graph by hand, we first sketch the three graphs y = - 1, y = x - 1, and y = x 2 , and then use the appropriate portion of each. x y - 1 1 1 Notice that solid dots indicate points on the graph, whereas the open dots indicate points not on the graph. For example, when x = 1, the inequalities in the formula tell us that we are to use the latest formula ( x 2 ) rather than the middle one ( x - 1). Thus f (1) = 1 2 = 1, not 0, so we place a solid dot at (1 , 1) and an open dot at (1 , 0).
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10 J. S´ anchez-Ortega Example 2.5. Given the graph of the function f -5 -4 -3 -2 -1 1 2 3 4 5 0 -4 -3 -2 -1 1 2 x y a. Find the technology formula of f .
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