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1.3 - New Functions From Old Functions

1.3 - New Functions From Old Functions. Translation : f ( x ) + k. Graph y 1 = x 2 on your graphing calculator and then graph y 2 given below to determine the movement of the graph of y 2 as compared to y 1 . Generalize the effect of k. Translation : f ( x – h ).

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1.3 - New Functions From Old Functions

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  1. 1.3 - New Functions From Old Functions

  2. Translation: f (x) + k Graph y1 = x2 on your graphing calculator and then graph y2 given below to determine the movement of the graph of y2 as compared to y1. Generalize the effect of k.

  3. Translation: f (x – h) Graph y1 = x2 on your graphing calculator and then graph y2 given below to determine the movement of the graph of y2 as compared to y1. Generalize the effect of h.

  4. Horizontal and Vertical Translationsf (x – h) + k • The value of h causes the graph of f(x) to translate left or right (horizontally). If h < 0, the graph shifts h units left. If h > 0, the graph shifts h units right. • The value of k causes the graph of f(x) to translate up or down (vertically). If k > 0, the graph shifts k units up. If k < 0 then the graph shifts k units down.

  5. Examples Use the library of functions to sketch a graph of each of the following without using your graphing calculator. • f(x) = (x + 3)2 – 2 • g(x) = | x – 2 | + 3

  6. Verify this on your graphing calculator by graphing: x- and y-Axis Reflections • The graph of y = - f(x) is the same as graph of f(x) but reflected about the x-axis. • The graph of y = f(-x) is the same as graph of f(x) but reflected about the y-axis.

  7. Compression and Stretches • The graph of y=af(x)is obtained from the graph of y = f(x)by vertically stretching the graph if |a| > 1 or vertically compressing the graph if 0 < |a| < 1. Verify this by graphing y1 = |x| y2 = 3|x| y2 = (⅓) |x|

  8. The Algebra of Functions

  9. The sum f + g is the function defined by (f + g)(x) = f(x) + g(x) The domain of f + g consists of numbers x that are in the domain of both f and g (the intersection of the domains). The difference f - g is the function defined by (f - g)(x) = f(x) - g(x) The domain of f - g consists of numbers x that are in the domain of both f and g (the intersection of the domains).

  10. The product f ∙g is the function defined by (f ∙ g)(x) = f(x) ∙ g(x) The domain of f ∙g consists of numbers x that are in the domain of both f and g(the intersection of the domains). The quotient f / g is the function The domain of f / g consists of all x such that x is in the domain of f and g and g(x)≠ 0 (the intersection of the domains).

  11. Composition of Functions Given two functions f and g, the composite function is defined by (f ◦ g)(x) = f(g(x)) Read “f composite g of x” The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. Note: In general (f ◦ g)(x) ≠ (g ◦ f)(x)

  12. Range of g (equal to or a subset of) Domain of g Domain of f Range of f g f x g(x) f(g(x)) Domain of f(g) Range of f(g) (f ◦ g)(x) = f(g(x)) Or f(g)

  13. Composition of Functions Let f(x) = 2x – 3 and g(x) = x 2 – 5x. Determine (f ◦ g)(x). (f ◦ g)(x) = f(g(x)) = f(x 2– 5x) x = 2 – 3 (x 2– 5x) = 2x 2 – 10x – 3

  14. Examples (a) Let and g(x) = x2 – 2. Determine (i) (f ◦ g)(x) and (ii) (g ◦ f)(x). Determine the domains of each. • If y = cos (x2 – 2) and y = (f ◦ g)(x), determine f and g. • If y = esin(x+5) and y = (f ◦ g ◦ h)(x), determine f, g, and h.

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