A is a function that can be written as a whose and are .
f(x)=polynomial/polynomial
We are asked to compare an and a vertical of a rational function. To do so, we will review the definitions and properties of excluded values and asymptotes of rational functions.
Excluded Values of Rational Functions
In general, an excluded value is a value for which a function is . A rational function is defined only for the values of the that do not make the denominator equal 0 and it is undefined for values of x that make the denominator 0. There are rational functions whose denominator will never be equal to 0. These functions do not have excluded values.
| Rational Function
|
Denominator =0
|
Excluded Values
|
| y=1/x-1
|
x-1=0
|
x=1
|
| y=x+1/x^2-2x+1
|
x^2-2x+1=0
|
x=1
|
| y=x/x^2-1
|
x^2-1=0
|
x=- 1 and x=1
|
| y=2x-1/x^2+1
|
x^2+1=0
|
None
|
Vertical Asymptotes of Rational Functions
An asymptote is a that the graph of a function gets closer to as x or y gets larger in . If the line is vertical it is a vertical asymptote. Here are a few examples of rational functions which have vertical asymptotes.
Note that not every rational function has a vertical asymptote.
A rational function is written in simplest form when its numerator and denominator have no common factors. When a rational function is written in simplest form, it has a vertical asymptote at each of its excluded values. If a function is not in simplest form, it can be simplified and then it will have a vertical asymptote at each excluded value.
| Rational Function
|
Simplest Form
|
Excluded Values of Simplest Form
|
Vertical Asymptotes
|
| y=1/x-1
|
y=1/x-1
|
x=1
|
x=1
|
| y=x+1/x^2-2x+1
|
y=x+1/x^2-2x+1
|
x=1
|
x=1
|
| y=x/x^2-1
|
y=x/x^2-1
|
x=- 1 and x=1
|
x=- 1 and x=1
|
| y=x/x^2+x
|
y=1/x+1
|
x=- 1
|
x=- 1
|
| y=2x-1/x^2+1
|
y=2x-1/x^2+1
|
None
|
None
|
| y=x^2-x+4/x^4+x^2+1
|
y=x^2-x+4/x^4+x^2+1
|
None
|
None
|
| y=x^3+3x/x^2+3
|
y=x
|
None
|
None
|
| y=x^2-4/x+2
|
y=x-2
|
None
|
None
|
Note that the excluded values of a rational function should always be identified using its original form. This is because the excluded values of the simplest form can be different from the excluded values of the original function.
| Rational Function
|
Excluded Values
|
Simplest Form
|
Excluded Values of Simplest Form
|
| y=1/x-1
|
x=1
|
y=1/x-1
|
x=1
|
| y=x+1/x^2-2x+1
|
x=1
|
y=x+1/x^2-2x+1
|
x=1
|
| y=x/x^2-1
|
x=- 1 and x=1
|
y=x/x^2-1
|
x=- 1 and x=1
|
| y=x/x^2+x
|
x=- 1 and x=0
|
y=1/x+1
|
x=- 1
|
| y=2x-1/x^2+1
|
None
|
y=2x-1/x^2+1
|
None
|
| y=x^2-x+4/x^4+x^2+1
|
None
|
y=x^2-x+4/x^4+x^2+1
|
None
|
| y=x^3+3x/x^2+3
|
None
|
y=x
|
None
|
| y=x^2-4/x+2
|
x=- 2
|
y=x-2
|
None
|
Conclusion
Let's gather the information about the excluded values and the vertical asymptotes of all the previously mentioned functions. We will also note whether each function is written in simplest form.
| Rational Function
|
Excluded Values
|
Vertical Asymptotes
|
Simplest Form?
|
| y=1/x-1
|
x=1
|
x=1
|
Yes ✓
|
| y=x+1/x^2-2x+1
|
x=1
|
x=1
|
Yes ✓
|
| y=x/x^2-1
|
x=- 1 and x=1
|
x=- 1 and x=1
|
Yes ✓
|
| y=x/x^2+x
|
x=- 1 and x=0
|
x=- 1
|
No *
|
| y=2x-1/x^2+1
|
None
|
None
|
Yes ✓
|
| y=x^2-x+4/x^4+x^2+1
|
None
|
None
|
Yes ✓
|
| y=x^3+3x/x^2+3
|
None
|
None
|
No *
|
| y=x^2-4/x+2
|
x=- 2
|
None
|
No *
|
Based on this table, we can make the following observations.
- If a rational function is written in simplest form, its excluded value is x= a its vertical asymptote is x= a.
- If a rational function is not written in simplest form, it has a vertical asymptote at x= a if and only if x= a is an excluded value of the simplest form of the function.
- If a rational function has no excluded values, it has no vertical asymptotes. It does not mater if it is written in simplest form.
In conclusion, each excluded value of the simplest form of a rational function corresponds to a vertical asymptote.
