# Domains of Rational Functions (standard notation)

• Shafty
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
Shafty
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.

{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.

You could use interval notation.

$$(-\infty, -5) \cup (-5,7) \cup (7,\infty)$$

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
$$\{x | x>-5 or -5<x< 7 or 7< x\}$$
or
$$\{x | x\ne -5 and x\ne 7\}$$
or
$$\{x| x< -5\}\cup \{x| -5< x< 7\}\cup \{x| x> 7\}$$

Last edited by a moderator:
I've never seen

$$-5 < x > 7$$

considered a proper inequality: I believe Halls has a typo and that center piece
should be $\{x | -5 < x < 7 \}$.

Last edited:
Thanks, I have corrected it. (And will now pretend I never wrote such a silly thing!)

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.

• General Math
Replies
6
Views
847
• General Math
Replies
4
Views
896
• General Math
Replies
16
Views
2K
• General Math
Replies
5
Views
950
• General Math
Replies
2
Views
1K
• General Math
Replies
8
Views
2K
• General Math
Replies
11
Views
11K
• General Math
Replies
4
Views
408
• General Math
Replies
1
Views
771
• General Math
Replies
2
Views
2K