| | | Domains of Rational Expressions

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined.

The denominator can not be zero because, nothing can be divided by zero and remain a real number.

The first letter of my names starts with a "T" so these are the rational expressions that I'll be solving:

1 - x^2 and 2b - 2 x 2b^2- 8

1 - x^2 This is the first given expression. Since we know that denominator is the valuable, x can not equal zero be because it will make the expression undefined, but any x other real number will solve the expression.

x = 0 The zero is the excluded value. We can see that the domain for this expression is a set of all real numbers where zero is excluded from the value.
The expression is written like this:D = {x| x € R b ≠ 0}.

For the second expression, I'll have to set the denominator equal to zero the find the excluded values.

2b - 2 Again, this is the given expression. In order for me to find the excluded values, I'll have to factor the denominator and make them equal to zero in order to solve for b.
2b^2 - 8

2b - 2 This is the denominator that has to be factored.

(2b + 4)(b - 2) The expression is now factored.

2b + 4 = 0 This is the first factor that is equal to zero.

2b = -4 Next I'll subtract 4 from both sides to isolate my valuable.

2b = -4 This expression is then divided by 2 to isolate b. 2 2

b = -2 -2 is one of the excluded valuables that will make the denominator equal zero.

(b - 2) = 0 This is the second factor that has to be set equal to zero to solve the excluded value.

b = 2 2 is added to both sides that will get the second expression to make the denominator equal to zero.

Since I have the terms 2 and - 2, once substituted for the valuable b, it will make the denominator equal to zero therefore excluding values in the domain.
This can be expressed as D = {b| b € R ≠ ± 2}.

There are excluded values in both expressions. A excluded value is a real number where the variable is substituted which will make the denominator in the expression equal to zero making it undefined.
Since x was alone in the first expression, and was the variable in the denominator, it can not be equal to zero, therefore making it the excluded value.
The second expression was a little different because 2 and -2 was substituted for the variable b, making the denominator equal to zero, making 2 and -2 the excluded variable.

Dugopolski, M (2012) Elementary and Intermediate Algebra, 4th Edition. McGraw Hill New York, New York | |