Question

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

a. n is even
b. n is odd
c. n < 0
d. n > 0

Answer 1

We are given function f(x)=x^n.

We need to determine value of n so that inverse of the given function is also represents a function.

Please note: A relation(an equation) is a function if each value of it's domain has exactly one value. On other words, there should not be two values of the function for each x value we take for function.

Let us try to find the inverse of the function now.

Let us replace f(x) by y first.

We get y=x^n.

Now, we need to solve it for x.

Taking nth root on both sides, we get

$\sqrt[n]{y}=\sqrt[n]{x^n}$

On simplifying, we get

$x=\sqrt[n]{y}$

Switching x and y, we get

$y=\sqrt[n]{x}$

We got a nth radical (x).

For an even radical we always get two different values (+ and -).

But for an odd radical we always get a single value.

Therefore, n should be an odd whole number.

So, the correct option is b. n is odd