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Intro to Functions Mr. Gonzalez Algebra 2
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Linear Function (Odd) Domain (- , ) Range (- , ) Increasing (- , ) Decreasing Never End Behavior As x , f(x) As x - , f(x) -
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Quadratic Function (Even) Domain (- , ) Range [0, ) Increasing (0, ) Decreasing (- , 0) End Behavior As x , f(x) As x - , f(x)
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Cubic Function (Odd) Domain (- , ) Range (- , ) Increasing (- , 0) (0, ) Decreasing Never End Behavior As x , f(x) As x - , f(x) -
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Absolute Value Function (Even) Domain (- , ) Range [0, ) Increasing (0, ) Decreasing (- , 0) End Behavior As x , f(x) As x - , f(x)
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Square Root Function (Neither) Domain [0, ) Range [0, ) Increasing (0, ) Decreasing Never End Behavior As x , f(x) As x - , f(x) 0
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Cube Root Function (Odd) Domain (- , ) Range (- , ) Increasing (- ,0) (0, ) Decreasing Never End Behavior As x , f(x) As x - , f(x) -
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Exponential Function (Neither) Domain (- , ) Range (0, ) Increasing (- , ) Decreasing Never End Behavior As x , f(x) As x - , f(x) 0
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Logarithmic Function (Neither) Domain (0, ) Range (- , ) Increasing (- , ) Decreasing Never End Behavior As x , f(x) As x - , f(x) -
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Inverse Function (Odd) Domain (- , 0) (0, ) Range (- , 0) (0, ) Increasing Never Decreasing (- ,0) (0, ) End Behavior As x , f(x) 0 As x - , f(x) 0
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Inverse Squared Function (Even) Domain (- , 0) (0, ) Range (0, ) Increasing (- ,0) Decreasing (0, ) End Behavior As x , f(x) 0 As x - , f(x) 0
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Constant Functions (Even/Neither) Horizontal Domain (- , ) Range (y) Vertical Domain (x) Range (- , )
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Step Function (Neither) Domain (- , ) Range (only integers) Increasing Never Decreasing Never End Behavior As x , f(x) As x - , f(x) -
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Maximum and Minimums (Extrema) Absolute Max and MinRelative Max and Min We will have an absolute maximum (or minimum) at provided f(x) is the largest (or smallest) value that the function will ever take on the domain that we are working on. There may be other values of x that we can actually plug into the function but have excluded them for some reason. A relative maximum or minimum is slightly different. All that’s required for a point to be a relative maximum or minimum is for that point to be a maximum or minimum in some interval of x’s around. There may be larger or smaller values of the function at some other place, but relative to, or local to, f(c) is larger or smaller than all the other function values that are near it.
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Example #1 Domain (- , ) Range (- , ) Increasing (- ,-2) (0, ) Decreasing (-2, 0) End Behavior As x , f(x) As x - , f(x) - Extrema Relative Max at y=3 Relative Min at y=-3
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Example #2 Domain (- , ) Range [-7, ) Increasing (-0.5, 1.5) (2.5, ) Decreasing (- , -0.5) (1.5, 2.5) End Behavior As x , f(x) As x - , f(x) Extrema Absolute Min at y=-7 Relative Max at y=1 Relative Min at y=-1.5
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Example #3 Domain (- , ) Range (- , 17] Increasing (- , -0.5) (0.5, 1.5) (2, 2.5) Decreasing (-0.5, 0.5) (1.5, 2) (2.5, ) End Behavior As x , f(x) - As x - , f(x) - Extrema Absolute Max at y=17 Relative Max at y=0.5 and y=1 Relative Min at y=-3.5 and y=0
![Example #3 Domain (- , ) Range (- , 17] Increasing (- , -0.5) (0.5, 1.5) (2, 2.5) Decreasing (-0.5, 0.5) (1.5, 2) (2.5, ) End Behavior As x , f(x) - As x - , f(x) - Extrema Absolute Max at y=17 Relative Max at y=0.5 and y=1 Relative Min at y=-3.5 and y=0](http://images.slideplayer.com./27/8983869/slides/slide_17.jpg "Example #3 Domain (- , ) Range (- , 17] Increasing (- , -0.5) (0.5, 1.5) (2, 2.5) Decreasing (-0.5, 0.5) (1.5, 2) (2.5, ) End Behavior As x , f(x) - As x - , f(x) - Extrema Absolute Max at y=17 Relative Max at y=0.5 and y=1 Relative Min at y=-3.5 and y=0")
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Example #4 Domain [-5, ) Range [-6, ) Increasing [-5, ) Decreasing Never End Behavior As x , f(x) As x - , f(x) -6 Extrema Absolute Min at y=-6
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Example #5 (Piecewise Functions) Domain (- , -1) [-1, ) or (- , ) Range [-3, ) Increasing (0, ) Decreasing [-1, 0) Constant (- , -1) End Behavior As x , f(x) As x - , f(x) 1 Extrema Absolute Min at y=-3 Relative Max at y=-2
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Example #6 (Piecewise Functions) Domain (- 1] (1, 2) [2, ) or (- , ) Range (- , ) Increasing (- , 1) (2, ) Decreasing (1, 2) End Behavior As x , f(x) As x - , f(x) - Extrema Relative Max at y=4 Relative Min at y=-2
![Example #6 (Piecewise Functions) Domain (- 1] (1, 2) [2, ) or (- , ) Range (- , ) Increasing (- , 1) (2, ) Decreasing (1, 2) End Behavior As x , f(x) As x - , f(x) - Extrema Relative Max at y=4 Relative Min at y=-2](http://images.slideplayer.com./27/8983869/slides/slide_20.jpg "Example #6 (Piecewise Functions) Domain (- 1] (1, 2) [2, ) or (- , ) Range (- , ) Increasing (- , 1) (2, ) Decreasing (1, 2) End Behavior As x , f(x) As x - , f(x) - Extrema Relative Max at y=4 Relative Min at y=-2")
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Example #7 Domain (- , ) Range [0, ) Increasing (-1, 0.5) (2, ) Decreasing (- , -1) (0.5, 2) End Behavior As x , f(x) As x - , f(x) Extrema Absolute Min at y=0 Relative Max at y=2
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Example #8 (Piecewise Functions) Domain (- , 2] (2, ) or (- , ) Range [0, ) Increasing (0, 2] Decreasing [- , 0) Constant (2, ) End Behavior As x , f(x) 3 As x - , f(x) Extrema Absolute Min at y=0 Relative Max at y=2
![Example #8 (Piecewise Functions) Domain (- , 2] (2, ) or (- , ) Range [0, ) Increasing (0, 2] Decreasing [- , 0) Constant (2, ) End Behavior As x , f(x) 3 As x - , f(x) Extrema Absolute Min at y=0 Relative Max at y=2](http://images.slideplayer.com./27/8983869/slides/slide_22.jpg "Example #8 (Piecewise Functions) Domain (- , 2] (2, ) or (- , ) Range [0, ) Increasing (0, 2] Decreasing [- , 0) Constant (2, ) End Behavior As x , f(x) 3 As x - , f(x) Extrema Absolute Min at y=0 Relative Max at y=2")
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