- #1
tahayassen
- 270
- 1
Let f(x)=(x)^(x)
f(-1)=(-1)^(-1)
=1/-1
=-1
But according to wolframalpha, f(-1) does not exist.
f(-1)=(-1)^(-1)
=1/-1
=-1
But according to wolframalpha, f(-1) does not exist.
G.I. said:Fruthermore I can rewrite the f(x) in this way:
f(x)=exp(x*ln(x)).
And ln(x<0) is undefined.
tahayassen said:Let f(x)=(x)^(x)
But according to wolframalpha, f(-1) does not exist.
Best Pokemon said:http://www.wolframalpha.com/input/?i=f(x)+=+x^x
If you look at the graph of the function itself (without plugging anything in) you will see that it shows that at x=-1 it is imaginary.
tahayassen said:Let f(x)=(x)^(x)
f(-1)=(-1)^(-1)
=1/-1
=-1
But according to wolframalpha, f(-1) does not exist.
I saw the graph on Wolfram Alpha. Really nice pretty one if you constrain the x variable:Millennial said:So why did I go through this? I wanted to show why Wolfram does not display the graph for [itex]x \leq 0[/itex].
Millennial said:The function is defined as a real value only in the positive reals and the negative integers. For negative reals, we get the ugly-looking answer [itex]\displaystyle \frac{1}{x(\cos(x\pi)+i\sin(x\pi))} = \frac{\cos(\pi x)-i\sin(\pi x)}{x}[/itex], which can't be simplified further.
When x is a positive number, the exponent x means multiplication by itself x times. However, when x is a negative number, the exponent x does not have a clear meaning. Additionally, when x is equal to 0, any nonzero number raised to the power of 0 is equal to 1. Therefore, the graph of x^x is undefined for x is less than or equal to 0 because the exponent x does not have a clear meaning for negative numbers and 0.
No, imaginary numbers cannot be used to define the graph of x^x for x is less than or equal to 0. Imaginary numbers are used to represent the square root of a negative number, but in this case, the issue is not with the square root, but with the exponent x. The concept of raising a number to a negative or 0 exponent is not defined for imaginary numbers.
Yes, there are real-world applications where the graph of x^x is undefined for x is less than or equal to 0. For example, in population growth models, the function P(t) = P0 * r^t is used, where P0 represents the initial population, r represents the growth rate, and t represents time. However, this model does not hold for negative values of t, as it would represent a negative time period.
Yes, there are ways to extend the graph of x^x to include values of x less than or equal to 0. One way is to use the concept of complex numbers, where the exponent x can be defined for negative and 0 values. Another way is to use the concept of limits, where the limit of x^x as x approaches 0 from the right is defined as 1, and the limit as x approaches 0 from the left is undefined.
No, the graph of x^x cannot be defined for x is equal to 0. This is because any nonzero number raised to the power of 0 is equal to 1, and therefore, the graph of x^x would be a constant line at y=1 for x=0, which does not fit the definition of a function.