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Domains of Rational Expressions
I shall proceed to base my work on the following rational expressions:
X2 – 25 and 37 2 k2- k – 30
Set of all the numbers that are permitted to substitute for variables in an expression is known as the domain. The denominator determines which numbers will be allowed and which ones won’t be.
A number is not divisible by zero in our real number system. This is why we cannot have the denominator as a zero since the answer will be undefined.
The denominator is a constant term in the first expression. It is impossible for 2 to have the same value as zero; therefore, the domain won’t have any excluded values. The domain (D) comprises of all real numbers such that:
D = {x| x ∈ ℜ} or D = ℜ.
I must set my denominators to zero for me to find the excluded values for k.
k2- k – 30 = 0 I can use -6 and 5 to factorize
(k + 5) (k – 6) = 0setting each factor to be equal to zero
K + 5 = o or k – 6 = 0subtract five and add six to both sides respectively
K = -5 0r k = 6these are the excluded values for my second expression
The domain for the expression k2- k – 30 is the set of all Reals excluding -5 and 6 represented using set notation as D = {w| w ∈ ℜ, w ≠ -5, w ≠ 6 }
The two expressions both lack excluded values. This is so in the first expression as a non-zero constant cannot magically transform into a zero. The second expression has two excluded values because they would cause the denominator to become zero if inserted in the variable’s place. This will make the whole expression undefined.
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