Sketching x^x (1 Viewer)

klee98

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u could implicitly differentiate to find the stat pts, or you can just plot them like x = 0/ y =1 and stuff.
(-2)^(-2) isn't 4
 

Carrotsticks

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How would i sketch x^x? Also, why is f(x) not defined for x<0 ? When (-2)^(-2) = 4?
Try x=-1/2 to see why negative x values can get nasty.

Also, to obtain a rough sketch for positive x, realise that you are sketching
 

InteGrand

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So it wouldn't be worth it to graph the negative x values as only those work?
It'd be quite dodgy to graph the negative x (if you're only graphing on the real plane). The function is going to be well-defined if you allow complex values.
 

InteGrand

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So it wouldn't be worth it to graph the negative x values as only those work?
If you were interested in plotting the function for negative x ( using complex values, you'd be able to produce a visualisable 3D graph with negative x on the negative x-axis, and you could use the other two axes to represent and .

Using certain results like Euler's Formula etc., one may show that , so on one axis, you could plot and on the other plot , for all real x < 0.

Or, you could plot the real and imaginary parts on the one graph, and just have a 2D plot: Wolfram does this for you here -> http://www.wolframalpha.com/input/?i=plot+x^x . (Note that there is a graph for the real part, and one for the imaginary part. You can compare these to the graphs of and here to see that they match (only of interest to us are the parts of the graphs for x < 0): http://www.wolframalpha.com/input/?i=plot+|x|^x+cos+(pi*x),+|x|^x+*sin(pi*x))

Edit: those formulas for the Real and Imaginary parts of f give the principal roots. So they won't give the real-valued cube root for , say, but a complex one.
 
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