Interval notation of function's domain question

In summary, the domain of the function f(x) is all real numbers except for x=5, and the intervals where the function is defined can be described using interval notation as ##(- \infty,-2] \cup [3,+\infty)##.
  • #1
late347
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Homework Statement


find domain
f(x)= ## \frac{\sqrt{x^2-x-6}}{x-5} ##

Homework Equations

The Attempt at a Solution


[/B]
Well, one sees that x ≠5 and then we further investigate the dividend portion where the under-the-root expression ≥ 0

We can graph the parabola, and find the zeroes. Then investigate the intervals and keep in mind also that x ≠ 5 for the original function.

I'm interested in how is this domain described in the interval notation for domain such as with the [ ] brackets

I confirmed by pen-and-paper results with wolfram alpha and the particular type of notation that wolfram alpha uses is as follows.

domain: { x ∈ ℝ : x>5 or 5>x≥3 or x≤-2}
 
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  • #2
late347 said:

Homework Statement


find domain
f(x)= ## \frac{\sqrt{x^2-x-6}}{x-5} ##

Homework Equations

The Attempt at a Solution


[/B]
Well, one sees that x ≠5 and then we further investigate the dividend portion where the under-the-root expression ≥ 0

We can graph the parabola, and find the zeroes. Then investigate the intervals and keep in mind also that x ≠ 5 for the original function.

I'm interested in how is this domain described in the interval notation for domain such as with the [ ] brackets

I confirmed by pen-and-paper results with wolfram alpha and the particular type of notation that wolfram alpha uses is as follows.

domain: { x ∈ ℝ : x>5 or 5>x≥3 or x≤-2}

Domain: ##(- \infty,-2] \cup [3,5) \cup (5, +\infty)##

Or: ##(- \infty,-2] \cup [3,+\infty) - \{5\}##
 
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1. What is interval notation?

Interval notation is a way to represent the domain of a function using intervals on a number line. It is written as [a, b] or (a, b), where a and b are the endpoints of the interval. Square brackets are used for inclusive endpoints, meaning that the endpoints are included in the interval, while parentheses are used for exclusive endpoints, meaning that the endpoints are not included in the interval.

2. How is interval notation used to represent a function's domain?

Interval notation can be used to represent a function's domain by indicating the set of all possible input values that the function can take. For example, if a function is defined on the interval [3, 8], it means that the function can take any value between 3 and 8, including 3 and 8.

3. What does it mean when an interval is unbounded?

An unbounded interval is one where at least one of the endpoints is either positive or negative infinity. In interval notation, this is represented by using the symbols -∞ (negative infinity) or ∞ (positive infinity). For example, the interval (2, ∞) represents all values greater than 2.

4. Can a function have multiple intervals in its domain?

Yes, a function can have multiple intervals in its domain. This means that there are multiple sets of input values that the function can take. For example, a function defined on the intervals [2, 5] and (8, 10) means that the function can take values between 2 and 5, as well as values greater than 8 and less than 10.

5. How do you determine the domain of a function using interval notation?

To determine the domain of a function using interval notation, first identify any excluded values, such as values that would result in division by zero or taking the square root of a negative number. Then, write the intervals for all possible input values, including any unbounded intervals. The union of these intervals represents the domain of the function.

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