Question

Determine the values of n for which f(x)=x" has an inverse that is a function. Assume that n is a whole number.

a. nis even
c.no
b. nis odd
d. => 0
Please select the best answer from the choices provided
O
A

Answer 1

Assuming you're asking "for which values of
`n` the function
`x^n` has an inverse that is a function", the answer is "all the odd exponents
`n`".

Infact, if
`n` is even, you have that

`x^n=(-x)^n \quad \forall x \in \mathbb{R}`

and so
`f(x)=x^n` is not injective, and thus not invertible

On the other hand, if
`n` is odd, we have:

• 
  `\lim_(x\to\pm\infty)x^n=\pm\infty`
• 
  `x^n` is continuous.
• The first two points tell us that the function is surjective.
• Moreover, the derivative is
  `f'(x)=nx^(n-1)`. Since
  `n-1` is even, we have
  `f'(x)>0`, thus the function is always increasing, and so the function is also injective.
• Injective and surjective means bijective, and the function can be inverted.

Answer 2

Answer:n is odd

Explanation: