limits on a rational expression

[C3PO]

When we talk about rational expressions, we also talk about limiting the domain or range of a function. How are functions limited when we create rational expressions such as 6/x? I understand that each input should be specifically related to an output and that you should omit any values that produce a zero denominator, but how does 6/x limit the function? I find functions and their related domain/range confusing. Thanks.
C3PO

 

[DrPhil]

When we talk about rational expressions, we also talk about limiting the domain or range of a function. How are functions limited when we create rational expressions such as 6/x? I understand that each input should be specifically related to an output and that you should omit any values that produce a zero denominator, but how does 6/x limit the function? I find functions and their related domain/range confusing. Thanks.
C3PO

You have said it: you can not allow any value of x that would lead to the denominator being zero. For the function f(x) = 6/x, the domain must exclude x=0. Any other value of x can be included in the domain, but the function simply does not exist when x=0. The range of the function is unlimited in this case .. ANY value of f(x) can be divided by 6 to give a value of x in its domain.

 

[C3PO]

Recall and practice

I am not doing homework yet, just attempting to recall "how to" because its been a while since I was in school. Dr. Phil you are awesome! Thanks for your fast reply last post. I have one more question for the day; how would you use factoring to find the LCD for the denominators of 1,2,3,4,5, 12 and 144? Thank you for your time and expertise.
C3PO

 

[HallsofIvy]

I am not doing homework yet, just attempting to recall "how to" because its been a while since I was in school. Dr. Phil you are awesome! Thanks for your fast reply last post. I have one more question for the day; how would you use factoring to find the LCD for the denominators of 1,2,3,4,5, 12 and 144? Thank you for your time and expertise.
C3PO

Start by factoring. You can ignore 1. Then 2, 3, and 5 cannot be factored any further- they are prime numbers.
4= 2(2), 12=3(4)=(2)(2)(3), and 144= (12)(12)= ((2)(2)(3))((2)(2)(3))= (2)(2)(2)(2)(3)(3).

The least common denominator (also "least common multiple") must have all the factors of any one of the numbers.
Notice that 12= 2(2)(2)(2)(3)(2) already has the factors of all previous numbers except 5- all of 2, 3, 4, and 12 divide 144. So we just need to multiply 144 by 5. The least common denominator is 144(5)= 720.

 

[mmm4444bot]

How are functions limited when we create rational expressions such as 6/x?

DrPhil told you how these function types are limited, where x causes the denominator to approach zero: their behavior is unlimited (i.e., function value approaches positive or negative infinity).

These functions are also limited, where the denominator becomes huge (in absolute value). In these parts of the domain, the function value approaches zero (from below or from above).