J(x)=x^-1 is an example of: ; and if so, why?

Question

A) an inverse function.

B) a linear function with negative slope.

C) a constantly diminishing function.

Please explain why and/or show 'worked' solution or online source. Thank you.

Answer 1

The first answer here is incorrect. J(x) = x^(-1) may be an example of a function that is its own inverse in that it's symmetric about y = x (switching x and y, y = 1/x --> x = 1/y --> y = 1/x), but it's not by definition an inverse function. An inverse function is noted as J^(-1)(x), in that it is the inverse of the function. Otherwise, if you are raising the variable to the power of -1, it's x^(-1) = 1/x.

So J(x) = 1/x. This is:

c) A constantly diminishing function.

As x is increasing, 1/x is decreasing. You're dividing 1 by a larger and larger number so J(x) must be decreasing with increasing x.  

Answer 2

a) because begative exponents are simplified by moving them to the opposite part of the fraction.  So x^-1 become 1/x^1 which is just 1/x.  1/x is the simplest/definitive inverse function, because if you make an equation y = 1/x, the y will decrease as x increases, and vice-versa.