which statement best describes the function f(x)=x^3-x^2-9x+9

Question

A) It is not a function
B) it is a one to one function
c) it is a many to one function
D) it fails the vertical line test 

Answer 1

You must analyze and graph or sketch this function to answer. This function has 3 roots, at x= 1, -3, and +3. This can be found by factoring first by (x-1) using synthetic division, resulting in a remainder of x^2-9, so it factors to:

 

(x-1)(x+3)(x-3)

 

This means there are multiple values of x for a given y, for a portion of the curve, so it is definitely not a one to one function. It actually passes the vertical line test, but not the horizontal line test, so the best answer is that it is a many to one function.

 

- Ben

Answer 2

Hello Kim,

 

Answers A and D are redundant in that if the equation had failed the vertical line test then it would not have been a function either.  So the only remaining answers are one to one function or many to one function.  A many to one function has one or more peak(s) and/or valley(s).

 

The function provided is a polynomial and has a peak and a valley.  There are a few ways to show this: Plug in x=-10, -5, 0, 5, 10, etc and compare ups and downs in the evaluations; Take the derivative and see that it is sometimes negative and sometimes positive; or Show that it has multiple roots as Ben did.  If a function has multiple roots then there must be a peak or valley between each.  Likewise, if the derivative shifts from negative to positive or vice versa then there's an up-slope and down-slope with a peak of valley between them.

 

The answer is C.  It is a many to one function.

 

Michael