Cosa succede se l'esponente in una funzione di potenza è negativo?
TLDR:
Versione lunga:
If the exponent of a power function is negative, you have two possibilities:
- the exponent is even
- the exponent is odd
The exponent is even:
#f(x) = x^(-n)# where #n# è anche.
Anything to the negative power, means the reciprocal of the power.
Questo diventa #f(x) = 1/x^n#.
Now let's look at what happens to this function, when x is negative (left of the y-axis)
The denominator becomes positive, since you're multiplying a negative number by itself an even amount of time. The smaller#x# is (more to the left), the higher the denominator will get. The higher the denominator gets, the smaller the result gets (since dividing by a big number gives you a small number i.e. #1/1000#).
So to the left, the function value will be very close to the x-axis (very small) and positive.
The closer the number is to #0# (like -0.0001), the higher the function value will be. So the function increases (exponentially).
What happens at 0?
Well, let's fill it in in the function:
#1/x^n = 1/0^n#
#0^n# è ancora #0#. You're dividing by zero! ERROR, ERROR, ERROR!!
In mathematics, it is not allowed to divide by zero. We declare that the function doesn't exist at 0.
#x=0# è un asintoto.
What happens when x is positive?
quando #x# è positivo, #1/x^n#, stays positive, it will be an exact mirror image of the left side of the function. We say the function is anche.
Mettere tutto insieme
Remember: we have established that the function is positive and increasing from the left side. That it doesn't exist when #x=0# and that the right side is a mirror image of the left side.
With these rules the function becomes:
What about an odd exponent?
The only change with an odd exponent, is that the left half becomes negative. It is mirrored horizontally. This function becomes:
Spero che questo ha aiutato!