Thread: Ranges without the aid of a calculator Help

This is a pre-calc review for Calculus (I think this is the right area)

I can do domains completely fine but I have trouble with ranges. Like for instance y=sqrt(x+1), y=1/(sqrt(x+1)) and y= sqrt((1/x)+1). For the first one, I know its a radical function and y >= 0. But Mathematically how do I calculate this? I heard you put the equation into the inverse function but I'm not really sure what to do afterward.

Thanks!

You don't need to calculate it for the first square root. If you need a justification you can start by looking at . This is a parabola centered at the origin and it opens to the right (towards positive x-values). So x can only have positive values. This is equivalent to . In your case you are taking the positive square root, which will be the part of the parabola above the x-axis, that is the square root evaluates to only value greater or equal to 0.

In your first case, the range is . In the second expression you are taking the reciprocal of the first function. The range is . Here is how you get it. Lets look at a point in the first range. To get the range of the second one, we take the reciprocal, i.e. . When y is close to 0 we get a big number, so on one side the range can be arbitrarily large. However, if we make really large, we are making really tiny, so the range is 0 at the other end. The range ought to be , but since you can never make zero, it must be an open interval -

Originally Posted by Nekuni

This is a pre-calc review for Calculus (I think this is the right area)

I can do domains completely fine but I have trouble with ranges. Like for instance y=sqrt(x+1), y=1/(sqrt(x+1)) and y= sqrt((1/x)+1). For the first one, I know its a radical function and y >= 0. But Mathematically how do I calculate this? I heard you put the equation into the inverse function but I'm not really sure what to do afterward.

Thanks!

Personally, I find the study of basic functions and transformations (vertical/horizontal stretch, shift up/down/left/right) to be useful when determining the domain and range of a function. For example, if you know the domain and range of x^2, then you can derive the domain/range list of 2x^2, (x-2)^2, and x^2 + 2 by realizing that these functions are simply products of transformations + their effect.

Also, here's a trick, (although it may be limited). you should be careful with it as some inverse functions can get quite messy.

We know that f: x -> y, where x is the domain and y is the range.
Now consider that the inverse function of f, call it g: y -> x

The domain of f is the range of g, and the range of f is the domain of g *verify*

For example, let f(x) = y = x^2
Domain: the real line
Range: y>=0

Now, g(x) = +- sqrt(x)
Domain: x>=0
Range: the real line

EDIT:// as an afterthought, if all else fails: graph it.