Domains of Rational Functions (standard notation)

In summary: The expression I wrote above IS what you really meant.In summary, the domain for this rational expression is all real numbers except for -5 and 7, which can be written in standard notation as (-\infty, -5) \cup (-5,7) \cup (7,\infty), with the alternative options of \{x | x>-5 or -5<x< 7 or 7< x\} or \{x | x\ne -5 and x\ne 7\} or \{x| x< -5\}\cup \{x| -5< x< 7\}\cup \{x| x> 7\}. It is important to note
  • #1
Shafty
3
0
Im preparing for a CLEP test in precalculus. As part of my prep, I need to review identifying domains of functions. I have a question about writing domains in standard notation. I was hoping someone could explain a bit the style.

For an example:

x-2 / x^2 -2x -35

As a rational expression, I know that the denominator can not be equal to zero. Therefore, to find the domain, I set the denominator equal to zero and solved the quadratic:

x = 7
x = -5

When x is either of these 2 values, the denominator will equal 0, and the expression is undefined. How would I write the domain in standard notation? I realize that the domain is all real numbers excluding -5 and 7, but is there a tidy way to write this?

Thanks.
 
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  • #2
{x | x<-5 U -5<x<7 U 7<x }

The domain is the union of the open intervals of x less than negative 5, x is greater negative 5 than but less than 7, and x is greater than 7.
 
  • #3
You could use interval notation.

[tex]
(-\infty, -5) \cup (-5,7) \cup (7,\infty)
[/tex]
 
  • #4
symbolipoint said:
{x | x<-5 U -5<x<7 U 7<x }

The domain is the union of the open intervals of x less than negative 5, x is greater negative 5 than but less than 7, and x is greater than 7.
I would NOT write it that way since the "U" notation is used for sets, not algebraic expressions. Either
[tex]\{x | x>-5 or -5<x< 7 or 7< x\}[/tex]
or
[tex]\{x | x\ne -5 and x\ne 7\}[/tex]
or
[tex]\{x| x< -5\}\cup \{x| -5< x< 7\}\cup \{x| x> 7\}[/tex]
 
Last edited by a moderator:
  • #5
I've never seen

[tex]
-5 < x > 7
[/tex]

considered a proper inequality: I believe Halls has a typo and that center piece
should be [itex] \{x | -5 < x < 7 \}[/itex].
 
Last edited:
  • #6
Thanks, I have corrected it. (And will now pretend I never wrote such a silly thing!)
 
  • #7
Shafty said:
For an example:

x-2 / x^2 -2x -35
As a side note, an expression such as this written on a single line should be written with parentheses around the numerator and denominator, like so:
(x-2) / (x^2 -2x -35)

Under the order of operations, the expression as you wrote it would be interpreted to mean
x - (2/x2) - 2x - 35, which I'm sure isn't what you really meant.
 

1. What is the standard notation for a rational function?

The standard notation for a rational function is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not equal to 0.

2. What is the domain of a rational function?

The domain of a rational function consists of all values of x for which the denominator, Q(x), is not equal to 0. This is because dividing by 0 is undefined in mathematics.

3. How do you find the domain of a rational function?

To find the domain of a rational function, set the denominator, Q(x), equal to 0 and solve for x. The values of x that make the denominator 0 will not be included in the domain. You may also need to consider any restrictions on the variable stated in the problem.

4. Can the domain of a rational function include non-real numbers?

No, the domain of a rational function can only include real numbers. This is because the standard notation for a rational function only allows for real numbers as inputs.

5. Are there any common mistakes to avoid when finding the domain of a rational function?

One common mistake to avoid is forgetting to check for any restrictions on the variable stated in the problem. Another mistake is incorrectly solving for x when setting the denominator equal to 0. It is also important to remember that the domain cannot include any values that make the denominator equal to 0.

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