A rational function is given. a. Complete each table for the function. b. Describe the behavior

step 1 So we have r of x is equal to x over x minus two. And we're looking at the characteristic of the

A rational function is given. a. Complete each table for the...

Key Concepts

Tabular Analysis of Function Behavior

Using tables to evaluate functions at selected input values is an effective method for visualizing the function's behavior. Tabular analysis helps in identifying patterns, such as the approach toward asymptotes and the rate of increase or decrease, which supports the understanding of the function’s local and end behavior.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They are central to understanding how functions behave when divided, and they often involve restrictions in their domain where the denominator is zero. Studying rational functions also includes analyzing their asymptotic behavior and limits.

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined due to the denominator equaling zero, and they indicate a tendency of the function to approach positive or negative infinity as the input nears those values. Understanding vertical asymptotes is essential when examining the behavior of rational functions near points of discontinuity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input grows large in the positive or negative direction. For rational functions, the degrees of the numerator and denominator determine the existence and location of a horizontal asymptote, which represents the value that the function approaches as the absolute value of the variable is taken to infinity.

Behavior Near Asymptotes

The examination of function behavior near asymptotes involves analyzing limits to determine how the function behaves on either side of a discontinuity or as the variable tends to infinity. This analysis reveals whether the function values increase without bound or approach a specific constant, and it is critical for understanding the overall behavior of a function.

Transcript

00:01 So we have r of x is equal to x over x minus two.

00:06 And we're looking at the characteristic of the vertical and horizontal asymptotes by doing some little bit of research.

00:12 So here's your table one.

00:14 And you have when x is 1 .5, 1 .9, 1 .99, 1 .999.

00:22 And we want to find what happens when we have that function.

00:27 And so one nice thing you can do is you can store into your calculator and you can type in like 1 .5 and then the stow button as x.

00:41 I'm going to do that right now, 1 .5 stow as x.

00:44 And then just hit enter.

00:46 And then you can type in to evaluate these expressions.

00:50 You can take x divided by left parentheses x minus two and just type it in.

00:57 And then i find out that this value for r of x is equal to negative 3.

01:02 When i go back, i can change my input as 1 .9 store as x.

01:09 And then i can just go back up and have that evaluate that expression.

01:13 Again, i get negative 19.

01:15 When i plug in 1 .99 for x and i do 1 .99 and either just plug it in here and figure out the value, or again, you can do the store and and then just go back up to your calculator and have it recalculate that expression, i get negative 199.

01:35 When i plug 1 .9999 in store that as x and then evaluate this expression, i end up getting negative 1 ,9999.

01:50 So i can see that as i get closer and closer to a value of 2, i'm going to negative infinity.

01:57 And then when we look at table 2, and we plug in some values that are to the right of 2, and plug in 2 .5, 2 .1, 2 .01, 2 .001.

02:12 Again, i can do exactly the same thing.

02:14 Take 2 .5 and store it as x.

02:16 And then i can go back on my calculator and have it do that calculation.

02:20 I get 5.

02:24 When i get closer, and i have 2 .1 stored as x and plug it in, i find out that that expression becomes 21.

02:34 And i'm not going to continue to go further and plug these other values in...

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