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
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Michael M. answered • 05/14/15
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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.
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