Find domain of (2+x-x^2)/((x-1)^2)

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In summary, the function (2+x-x^2)/((x-1)^2) has a domain of all real numbers except x = 1. It has a vertical asymptote at x = 1 and a horizontal asymptote at y = -1. The critical numbers are x = 5 and x = 1, with x = 5 being a stationary point and potentially a local minimum. The derivative of the function is (x-5)/(x-1)^3. To determine if the function is increasing or decreasing, we need to check the sign of the derivative. The function is increasing to the left of x = -1 and decreasing to the right of x = -1. It is also important to note
  • #1
bobboxx
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Could someone find the domain, intercepts, asymptotes, critical numbers, local min and max, absolute min and max, concavity, and inflection points of the function: (2+x-x^2)/((x-1)^2)

Thank you it would be much appreciated
 
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  • #2
We help out here, not do things for you. What do you know about these or what have you done so far?
 
  • #3
I have found a critical number at x=5 but I don't know if that is correct.

I have a vertical asymptote at X=1 and a horizontal at y=-1

My derivative of the function is (x-5)/(x-1)^3 but I am not sure f this is right either.

I don't know which points to use to tell if it is increasing or decreasing-do I use five and 1?
 
  • #4
bobboxx said:
I have found a critical number at x=5 but I don't know if that is correct.
Since x = 5 is a zero of the derivative, so f'(5) = 0, it is indeed a stationary point. You still need to check whether this is an extremum (and then max or min) or not.

bobboxx said:
I have a vertical asymptote at X=1 and a horizontal at y=-1
Both correct.

bobboxx said:
My derivative of the function is (x-5)/(x-1)^3 but I am not sure f this is right either.
The derivative is correct as well.

bobboxx said:
I don't know which points to use to tell if it is increasing or decreasing-do I use five and 1?
A function is rising where its derivitive is positive, a function is decreasing where the derivative is negative. So check the sign of f'(x).
 
  • #5
f'(5) is neither a minimum or a maximum

Why is this a critical number? The graph is increasing to the left of -1 and decreasing to the right of -1. The graph is not changing at 5.
 
  • #6
Since you derivative f'(x) = (x-5)/(x-1)^3, x = 5 is a zero of f'(x) so f'(5) = 0, therefore x = 5 is a stationary point which means the tangent line is parallel to the x-axis there. This may be a minimum, maximum or a point of inflection.

Do you know what it is?
 
  • #7
I think I figured it out.
f(5) would be a local minimum

I think I messed up because I choose f(0) as a point to the left of f(5) instead of a point in between 1 and 5.

Do you have to include your vertical asymptotes when you are trying to find out if the function is increasing or decreasing?

If yes, this is where I messed up. Thanks for your help.
 
  • #8
There is indeed a minimum at x = 5, that's right :smile:

Well, I don't know what you have to do but when we had to find vertical asymptotes, we also had to check whether the function approached infinity or -infinity to the left and right of the asymptote.
 
  • #9
Thanks appreciate it
 
  • #10
You're welcome, if there's anything else - do ask :smile:
 
  • #11
TD said:
You're welcome, if there's anything else - do ask :smile:

Yes, but ask in the Homework section of this site, which is located at the top.

The Math section is not for homework questions, as is clearly explained in the notice posted there entitled "Homework".
 

1. What is the definition of a domain?

The domain of a function is the set of all possible input values for which the function is defined.

2. How do I find the domain of a function algebraically?

To find the domain algebraically, look for any restrictions on the input values. In this case, the denominator cannot equal 0, so the domain is all real numbers except for 1.

3. Can I graph the function to determine the domain?

Yes, graphing the function can also help determine the domain. If there are any breaks or gaps in the graph, those points are not in the domain.

4. What happens if the domain is not specified?

If the domain is not specified, it is assumed to be all real numbers for which the function is defined.

5. Does the domain affect the range of a function?

Yes, the domain can affect the range of a function. If the domain is restricted, it may limit the possible output values or create gaps in the range.

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