Finding the Range of a Function

First of all, the domains you list on the last paragraph are wrong (unless you are missing parentheis). The domains for both of these problems are (x is all real numbers except zero).

To find the range, you need to understand the function. There is no mechanical way to do this.

Best to sketch a graph, and you need to know how to sketch a graph without a calculator.

Let's start with the first one. The first thing I notice is that it is a standard parabola X^2 - x -4 with an addition term 12/x subtracted from it.

I think the first thing I would do is work with the parabola. Where is the vertex? Does the parabola point up or down?

Then I would think about what the function 12/x looks like. I know it approaches infinity as x approaches zero. A very high or very low (negative) number makes this term insignificant.

Once you understand these two parts of the function, sketching the graph is fairly straightforward.

My guess is that the range will be between some maximum or minimum and either positive or negative infinity.

The key is to not look for a mechanical technique. Each of these problems is different. Look for a way to understand the function. Often separating it into functions you already understand is a good idea.

I hope this helps, I will leave the excersize for you. If you need additional help, please ask.

I will be happy to confirm your answer when you get it.

(Oh, should the real problem be ?

1. x^2 - x - 12/(x -4)

Those parenthesis are very important. If this is the case I would look at the parabola x^2 -x and the term 12/(x-4).

Good Look

this is what he meant, which is why he excluded 4 and 6 from the domain

there's no magic formula for range, you just need to look at the basic type of equation. a good tip is to try pluggin in infinity and negative infinity and zero and see what you get.

Stuh,

If you are right about the parenthesis, these both turn into linear equations.

In general I don't like your advice of pluggin' in values when these functions are easy to understand. It is a better strategy to try to figure out what the functions are about. In this case your strategy misses the whole point (and will not get the correct answer).

In the first case the function is f(x) = (x+3) when x is not equal to 4. This is true because the numerator is the quadratic function (x + 3) (x -4). This means that the range is all real numbers not equal to 7.

I will leave the second, as they say, as an exercise for the reader.

the suggestions to plug in infinity and negative infinity may not be particularly applicable to this example, but i think they can be helpful to get an idea of what kind of function it is in certain examples