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    Graphing of functions: f(x) = |x|^x

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    Use Graphing Utilities to examine a function of x : f(x) = |x|^x Lots of analysis of grpah of f(x)=|x|^x so please inspect the attachment for full treatise

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    Lots of graphs and explanations so look at the attachment as I cannot paste the graphs here.

    (a) We graph f(x) using the graphing utility [1] for the range -3 ≤ x ≤ 3, -2 ≤ y ≤ 2

    The settings to do this are shown in window below

    The resultant graphed function is then reproduced below

    Graphing of the function f(x) = |x|x in the range -3 ≤ x ≤ 3, -2 ≤ y ≤ 2

    Domain of f(x) is the range for all real valid x and can be seen to be from
    -infinity to + infinity

    (b) We use the zoom facility by examining the graph and zooming in on the area of the graph around (0,1). I have achieved this by simply altering the x range so that x extends from -0.2 to +0.2 and y extends from 0.0 to +2.0 using the graphing tools limit values. The zoomed graph can be seen below

    We can see that the limit of f(x) = |x|x as x tends to 0 is 1

    (c) The function f(x) = |x|x is continuous for all real numbers because there exists a real value of f(x) for all values of x; even at x=0 f(x) is defined. Graphically speaking this means that the curve of f(x) is unbroken at every point for real x

    (d) ...

    Solution Summary

    A graphing utility is used to graph the function f(x) in the range -3<=x<=3, -2<=f(x)<=2

    f(x) = |x|^x for x not = 0

    f(x) = 1 for x = 0

    The domain of f(x) is then determined

    Zoom and trace functions of the utility are then used to determine the limit of f(x) as x tends to zero

    The reasons why f(x) is continuous for all real numbers is then explained.

    From the graph the slope at (0,1) is then visually estimated