Cosa succede se l'esponente in una funzione di potenza è negativo?


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:
inserisci qui la fonte dell'immagine

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:
inserisci qui la fonte dell'immagine

Spero che questo ha aiutato!

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