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JuanSolo
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Homework Statement
What is the domain of f(x,y)=x^y
The Attempt at a Solution
I thought it would be all real pairs except (0,0)
but it is x>0 and y all real numbers?
JuanSolo said:Homework Statement
What is the domain of f(x,y)=x^y
The Attempt at a Solution
I thought it would be all real pairs except (0,0)
but it is x>0 and y all real numbers?
Vorde said:If ##z=x^y##, then ##ln(z)=yln(x)##
Do you see why x cannot be less than zero?
How about when x < 0, if y = 1/2 or y = √(2). What then?JuanSolo said:Sorry, I don't think I fully understand
Would it not be defined when y is something like 2 or whatever, even if x is negative?
It depends on your definition of exponentiation. If you were to restrict y to the integers, then yes, you could consistently define xy for negative values of x. On the other hand, for real numbers x and y, xy is often defined by xy=ey log x, which isn't defined if x<0.JuanSolo said:Sorry, I don't think I fully understand
Would it not be defined when y is something like 2 or whatever, even if x is negative?
Your line,loy said:but as long as x>0 , y can be defined in any value in R.
given x>0,
IF y>0 , f(x,y)=x^y .
IF y<0 , f(x,y)=x^(-y) = 1/(x^y)
IF y=0 , f(x,y)=1.
BUT when x<0 , the function is not defined , meaning , you can't get any answer, the function is discontinuous in x<0.
the domain of x is (0,∞) ,while y is (-∞,∞)
The domain of this function includes all real values for x and y, except for the point (0,0). This means that any real number can be raised to any real power, as long as it is not the base or exponent of 0.
The point (0,0) is excluded from the domain because it results in an undefined value for the function. When x=0, any value raised to the power of 0 is equal to 1, but when both x and y are 0, the function becomes 0^0 which is undefined.
No, the domain cannot be extended to include the point (0,0) because it would result in an undefined value for the function. Some mathematicians believe that 0^0 should be defined as 1, but this is still a topic of debate and is not universally accepted.
The domain of this function can be visualized on a 2-dimensional graph, with the x-axis representing the base (x) and the y-axis representing the exponent (y). The entire graph would be shaded, except for the point (0,0) which would be left blank.
No, there are no other restrictions on the domain of this function. As long as the point (0,0) is excluded, any other real number can be used as the base and exponent of the function.