Important Observations – Rational Function Exploration
Examine the rational function graphed below.
1. What value of x would create a zero denominator? Look on the graph, you will see that it is x = 1.
2. What happens to the y values as the x values approach x = 1 from the left side of the graph? As you look from left to right, you will see that y values decrease, that is, the graph turns downward. In fact, the y values continue to decrease forever as the graph gets closer and closer to x = 1.
3. What happens to the y values as the x values approach x = 1 from the right side of the graph? As you look from right to left, you will see that y values increase, that is, the graph turns upward. In fact, the y values continue to increase forever as the graph gets closer and closer to x = 1.
(Recall from college algebra: the imagined vertical line, x = 1, that this graph seems to get very close to, is called a vertical asymptote.)
4. Write out the domain and range of the function. The domain is relatively easy: the domain is all real numbers except x =1.
The range is a little harder: it looks like most all of y values are on the graph, but is every y value on the graph? There seems to be a y value that the graph gets close to also, that is, it seems like there is a horizontal asymptote, like perhaps y = 1, but is there?
You will learn more about this in chapter 2 when you study limits, but right now we can tell you that there is a horizontal asymptote: y = 1. So the range is all output values y except y = 1. You can verify this algebraically. Set y = 1 and see what happens in the equation.
next, multiply both sides by the denominator (x – 1)
(x – 1) = x
Now think about this for a minute. When is a number equal to itself minus 1?
Never! So you can see that this value of y (y = 1) leads to an absurd (untrue) statement.
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Highlight and type over the number 1 in the denominator and change it to some other number. What happens to the graph around that number? Repeat this for other values of the constant. x = 1.
The imagined line (asymptote) shifts to the number you type in.
Write out the domain and range of the function.
The domain is all x except the number you typed in the denominator. For example, if you typed in a 5, you have , and the domain is all real x except x = 5. The range is still all y values except y = 1.
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Increase the numerator in the same manner. Does this affect the domain? No. for example, if you typed in a 3 (so you had ) then the domain is still all x except x = 1.
Does it affect the range? Yes. The imagined horizontal line shifts up to y = 3, so the range is all y values except y = 3.
Don’t worry overmuch about the range in rational functions yet. You will understand this much better after we study limits in chapter 2. This is just a sort of “warm up” to the idea.