Do not try to find range algebraically!  As my example in Lesson 1, question 3 demonstrates, you can still miss key facts about the range even if you do your algebra correctly.  A safer way to find the range of a function is to draw a graph of the function and then visualize the range that way.

• Find the domain of the function first.  As I discuss in Lesson 1 of my book, the key to finding domain is identifying bottom zeros and regions that cause square root of a negative.
  
• Put the endpoints of the domain on a Table of Values and add about three more points in between the endpoints as well.

For example, if I were doing the range of my Lesson 1, question 3, I would do this:

• Make a table of values.  (Recall the question is f(x) = square root of "9-x^2".)  See the free sample of my book above, if you don't have my book.
• Since the domain is [-3,3] here, I would put -3, -2, -1, 0, 1, 2, and 3 in the x column of my table and compute the y-values.  Just make rough estimates of complicated y-values.  For example, when x= -2, you would get y= square root of 5, which is 2 and a bit, call it 2.2 (the exact result isn't important).  When x= -1, you get y= square root of 8, which is 2 and a bit more, almost 3, since square root of 9 is 3, call it 2.8 or something.
• Plot these points on your graph, and connect the dots from left to right.  You should see it makes a semicircle shape.  The range is now quite easy to read off.
• Just read the range starting from the lowest y value at the bottom of the graph and rising up to the highest y value at the top.  It is quite obvious in this case that the range is [0,3].
  

Make sure you have read the asymptote section in Lesson 2 (leading into question 15).  If the domain includes infinity or negative infinity, compute the limit as x approaches infinity and negative infinity to see what is happening to the function (and so y) at that time.

• If the limit as x approaches infinity is a finite value, let's say 2, for example, then you have a horizontal asymptote at y=2 on the far right of the graph.  That means as you draw the graph, it will come closer and closer to a horizontal line at y=2 on the far right of your graph, but never touch that line.  The other points you have plotted will help you establish whether the curve is above the horizontal asymptote or below it.
• If the limit as x approaches negative infinity is a finite value, let's say 3 , for example, then you have a horizontal asymptote at y=3 on the far left of the graph.  That means as you draw the graph, it will come closer and closer to a horizontal line at y=3 on the far left of your graph, but never touch that line.  The other points you have plotted will help you establish whether the curve is above the horizontal asymptote or below it.
• Note that the limits as x approaches positive infinity and negative infinity do not necessarily have to be the same!  They are two separate investigations looking for two separate horizontal asymptotes.
• A curve can cross a horizontal asymptote!  Do not think of a horizontal asymptote as a barrier.  Yes, at the far left of the graph, and far right, the curve will come and closer and closer to the asymptote but never touch.  However, in between there is nothing wrong with crossing a horizontal asymptote if you need to get from a point plotted below the asymptote up to a point plotted above it (or vice-versa).
  

If your domain has an endpoint that causes a bottom zero, it is possible that you have a vertical asymptote as x approaches that number on your graph.  Compute the limits as x approaches that value from the left side and the right side to see what the graph is up to.

• If the bottom zero causes k/0, you know that the limits are tending to infinity or negative infinity, causing a vertical asymptote.  The sign of the infinity tells you if the graph is sweeping up to infinity, or down to negative infinity as it approaches the asymptote from each side.
• If the bottom zero causes 0/0, you must factor and cancel to solve the limit.  If you get a finite answer, then you have found the x and y coordinates of a hole to plot on your graph.  Occasionally, after you factor and cancel, the limit transforms into a k/0 limit, so you are back to a vertical asymptote limit like I discuss above.
• Plot all those points and draw in any asymptotes you have found, and connect the dots to get a reasonably reliable picture of the function.

Once you have made your table-of-values, computed your limits as x approaches infinity and negative infinity (if necessary), and computed your limits as x approaches any value that causes a bottom zero, you are ready to sketch your graph.

Plot your points, and connect your dots from left-to-right.  Remember, a curve cannot cross a vertical asymptote, but a curve can cross a horizontal asymptote, if that is necessary to travel from one point to the next.

Once your sketch is drawn, read your graph from the very bottom (where y is negative infinity) to the very top (where y is positive infinity) to determine which y values are in the range.