Interval notation to indicate domain

Question

Use interval notation to indicate the domain of f(X):

 

Let f(x) = (x+5)/(x^2+4x+4)

 

Please help!! thank you. 

Answer 1

ANSWER:

 

f(x) is only restricted where x = -2, but all other x are in the domain of the function. Interval Notation is discussed below. Excluding only the exact point where x = -2, the domain should be written as (-∞,-2) ∪ (-2,+∞), where the union notation represents the joining of two intervals (one on each side of x = -2).

 

DOMAIN:

 

The domain of a function f(x) is only restricted where x will result in a value for f(x) which cannot be displayed on a Real Number graph of (x,f(x)). For this function, the denominator factors to (x+2)(x+2). Since a zero denominator will result when x+2=0 at x= -2, the domain cannot contain the point where x= -2. This would result in f(x) = 3/0, which is irrational. In the numerator, 5 can be added to any x and result in a Real Number. There are few cases where the numerator limits the domain. Any x value which will result in a square root of a negative number is one exception.

 

INTERVAL NOTATION:

 

Interval notation here needs to exclude only the exact point x= -2.

Interval notation is represented by brackets [ and ] and also by parenthesis ( and ).

Brackets indicate closed intervals which include one or more of the endpoints: [4,8], (-∞,5], or [7,+∞). 

By definition, infinity +∞ or negative infinity -∞ cannot be endpoints since they increase or decrease without bound. Parenthesis are always used with open intervals: where the exact endpoints are not included: (5,6), (-∞,6], or (-∞,+∞). 

The notation (-∞,+∞) shows no restriction or constraint on the domain. It represents all real numbers.

 

CHALLENGE QUESTION:

 

Express the domain of √(4-x) / (x² - 3x - 4):      Answer: (-∞,-1) ∪ (-1,4)