Finding Range of Undefined Function Algebraically: -1 / (x^2 + 3x +2)

[Onigma]

I was asked to find the range of -1 / (x^2 + 3x +2), entirely algebraically.

I began with finding the domain which turns out to be x cannot =-1 or -2 (after factorising the quadratic).

However, as I began to attempt to find the range, I was stuck. Creating a graph of the function makes it fairly easy to find the range, but I need to know is how do you find the range algebraically.

Thanks

 

[Dr.Peterson]

I was asked to find the range of -1 / (x^2 + 3x +2), entirely algebraically.

I began with finding the domain which turns out to be x cannot =-1 or -2 (after factorising the quadratic).

However, as I began to attempt to find the range, I was stuck. Creating a graph of the function makes it fairly easy to find the range, but I need to know is how do you find the range algebraically.

Thanks

One thing you can try is to find the inverse (which may be a relation rather than a function); the range of the given function will be the same as the domain of this inverse relation. (Hint: you'll likely be looking at a discriminant.)

 

[JeffM]

I was asked to find the range of -1 / (x^2 + 3x +2), entirely algebraically.

I began with finding the domain which turns out to be x cannot =-1 or -2 (after factorising the quadratic).

However, as I began to attempt to find the range, I was stuck. Creating a graph of the function makes it fairly easy to find the range, but I need to know is how do you find the range algebraically.

Thanks

There are at least two ways.

One is to ask yourself a few questions about f(x).

Are there values of x where f(x) is positive? Can you demonstrate the answer algebraically?

Is there a maximum positive value for f(x)? Can you demonstrate the answer algebraically?

Is there a minimum positive value for f(x)? Can you demonstrate the answer algebraically?

Are there values of x where f(x) is negative? Can you demonstrate the answer algebraically?

Is there a maximum negative value for f(x)? Can you demonstrate the answer algebraically?

Is there a minimum negative value for f(x)? Can you demonstrate the answer algebraically?

Does f(x) have any zeroes? Can you demonstrate algebraically?

Those are very obvious questions, but, for differentiable functions, it helps to know differential calculus to answer them. So another way to go is to find the domain of the inverse function (or the domains of the inverse functions if there is not a single inverse).

EDIT: I see that Dr. Peterson has beat me to the punch. I agree with him that for this function working with inverses is probably the easier way to go, and it definitely is the easier if you know no differential calculus. I am not sure, however, that working with inverse functions is always the easier way to go (nor am I implying thatDr. Peterson made any such assertion). Moreover, when working with inverses, remember that a function may have different inverses in different intervals of the domain.

SECOND EDIT: I also see that Dr. Peterson and I are using different vocabularies about inverses. He is talking about an inverse relation whereas I am talking about inverse functions in intervals. I do not believe that we are disagreeing about something fundamental, merely expressing the same fact in different words.

 

[Steven G]

I was asked to find the range of -1 / (x^2 + 3x +2), entirely algebraically.

I began with finding the domain which turns out to be x cannot =-1 or -2 (after factorising the quadratic).

However, as I began to attempt to find the range, I was stuck. Creating a graph of the function makes it fairly easy to find the range, but I need to know is how do you find the range algebraically.

Thanks

I would set -1 / (x^2 + 3x +2) = k and solve for x. If there is any restriction on x it will be in terms of k. So this will tell you which values of k you can not have for the range.

 

[Dr.Peterson]

I would set -1 / (x^2 + 3x +2) = k and solve for x. If there is any restriction on x it will be in terms of k. So this will tell you which values of k you can not have for the range.

This is equivalent to what we have said about inverses, but is perhaps easier to understand if inverses of functions that are not one-to-one are too unfamiliar. I recommend thinking this way.

 

[JeffM]

I would set -1 / (x^2 + 3x +2) = k and solve for x. If there is any restriction on x it will be in terms of k. So this will tell you which values of k you can not have for the range.

I like it.

I am not sure that this will work for really involved functions, but I am also quite sure that it is unfair to ask this question about really involved functions to students who do not know calculus.