I have a question about the rational function I recently posted. Would the range still be (-∞,2)U(2,∞) if the rational function is what Reiny posted "(2x^2-18)/(x^2+3x-10)." The y-intercept confuses me because its (0,1.8) and when I look at the line it passes through the horizontal asymptote.
Here is a wonderful webpage that let's your graph just about any curve
http://rechneronline.de/function-graphs/
enter your function with brackets in the form
(2x^2-18)/(x^2+3x-10)
set: "range x-axis from" -20 to 20
set: "range y-axis from" -20 to 20
you will see your y-intercept correct at (0,1.8)
and the two vertical asymptotes of x = -5 and x = 2
starting to show.
Horizontal asymtotes begin to show up only when x approaches ± infinity, so if you look to the far right, the curve approaches y = 2 from the bottom up, and if you look far to the left, the curve approaches y - 2 from the top down
It is very common for the curve to intersect the horizontal asymptote for reasonable small values of x.
If we set our function equal to 2,
(2x^2-18)/(x^2+3x-10) = 2 , and cross-multiply we get
2x^2 - 18x = 2x^2 + 6x - 20
-24x =-20
x = 20/24 = 5/6 , which is shown on the graph
But as x --> ±∞ , the function will never again reach the value of 2
(try it on your calculator, set x = 500 and evaluate
then let x = -500 and evaluate,
in the first case you should get 1.988... a bit below 2
in the 2nd case you should get 2.012... a bit above 2
the larger you make your x, the closer you will get to 2, but you will never reach it, and that is your concept of an asympote )
Thank you very much!
you are welcome,
does it make sense now?
Yes it does ^^
To determine the range of a rational function, it is important to consider the behavior of the function as the input values approach infinity and negative infinity.
In the case of the rational function (2x^2-18)/(x^2+3x-10), we can start by analyzing the horizontal asymptotes and the behavior of the function towards infinity.
1. Horizontal Asymptotes:
To find the horizontal asymptotes of a rational function, compare the degrees of the numerator and the denominator.
In this case, the degrees of the numerator and denominator are both 2. So, we need to compare the coefficients of the highest power of x in the numerator and denominator.
The coefficient of x^2 in the numerator is 2, and the coefficient of x^2 in the denominator is 1. Since the degree of the numerator and denominator are the same, the ratio of the coefficients (2/1) gives us the horizontal asymptote, which is y = 2.
2. Behavior Towards Infinity:
To determine the range of the function, we need to consider what happens to the function as x approaches positive and negative infinity.
As x approaches infinity, the terms involving x^2 become dominant in the numerator and denominator. So, the function behaves as if it is (2x^2/x^2) = 2. Therefore, as x approaches positive infinity, the function approaches the horizontal asymptote y = 2.
As x approaches negative infinity, the terms involving x^2 become dominant as well. So, the function behaves as if it is (-2x^2/-x^2) = 2. Hence, as x approaches negative infinity, the function also approaches the horizontal asymptote y = 2.
3. Range:
Since the function approaches y = 2 as x approaches both positive and negative infinity, the range of the function is (-∞, 2) U (2, ∞). This means that the function can take any value less than 2 or greater than 2, excluding the value 2 itself.
Regarding the y-intercept, its coordinates are (0, 1.8). The y-intercept doesn't affect the range of the function. The range is determined by the behavior of the function as x approaches positive and negative infinity, considering the horizontal asymptote.
I hope this explanation helps you understand why the range of the rational function (2x^2-18)/(x^2+3x-10) is (-∞, 2) U (2, ∞).