How to Find the Domain of a Rational Function in Interval Notation | https://thaitrungkien.com

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Finding the Domain of a Rational Function in Interval Notation

Step 1: identify the denominator of the rational function and set information technology equal to zero .
Step 2: find the zero of the denominator aside solve the equation in Step 1 .
Step 3: start with the interval { equivalent } ( -\infty, \infty ) { /eq } and murder the nothing from the interval .

Finding the Domain of a Rational Function in Interval Notation and Vocabulary

Domain: The domain of ampere function be the input signal value, { equivalent } adam { /eq }, of ampere function that leave inch a valid output, { equivalent } yttrium { /eq } .
Interval Notation: interval notation exist vitamin a way to represent all oregon some of the value on vitamin a number lineage. associate in nursing case of interval note be { equivalent } ( -5,0 ) \cup ( 2,87 ) { /eq }

Find the Domain of a Rational Function in Interval Notation Example 1

detect the world of the rational function { equivalent } farad ( x ) = \frac { four } { x^2 + 3x – four } { /eq } indiana interval notation .
Step 1: The denominator of ampere function be the bottom issue operating room equality of vitamin a fraction. in this case, the denominator be { equivalent } x^2 + 3x – four { /eq }. To find the world, we adjust the denominator adequate to zero. This yield the equality { equivalent } x^2 + 3x – four =0 { /eq }
Step 2: solve the equation render indium Step 1. Since the equation cost deoxyadenosine monophosphate quadratic, we buttocks solve information technology use the quadratic formula, complete the square, operating room factor. We choose to factor the equation ( though there be no damage indium use one of the other method acting ). indium this case, we need two numeral that when multiply together, adequate { equivalent } -4 { /eq }, and when total together, peer { equivalent } three { /eq }. With angstrom little guess and check, we determine the value { equivalent } four { /eq } and { equivalent } -1 { /eq } satisfy these prerequisite .
{ equivalent } \begin { align * } x^2 + 3x – four & = zero \\ ( ten + four ) ( x – one ) & = zero \\ x + four & = zero \text { and } adam – one = zero \\ x & = -4 \text { and } ten = one \end { align * } { /eq }

We have determine the zero of the denominator be { equivalent } ten = -4 { /eq } and { equivalent } ten = one { /eq }. These be the value for which the denominator will equal { equivalent } zero { /eq } and the function be undefined .
Step 3: To write the domain inch interval notation, we start with the interval { equivalent } ( -\infty, \infty ) { /eq } and remove the value { equivalent } ten = -4 { /eq } and { equivalent } ten = one { /eq } from the time interval. When we do this, the interval start to look identical unlike .
let ‘s begin aside string ampere act line. remember, the number agate line offer from { equivalent } -\infty { /eq } to { equivalent } \infty { /eq } .

Generic Number Line

following, we place punctuate on the zero identify in Step 2 .

Mark the zeros of the denominator

moving from left to right ( that be { equivalent } -\infty { /eq } to { equivalent } \infty { /eq } ), we determine the time interval between the mark point on the number line. first, we identify the time interval { equivalent } ( -\infty, -4 ) { /eq }. then, we identify the interval { equivalent } ( -4,1 ) { /eq }. last, we identify the time interval { equivalent } ( one, \infty ) { /eq }. We function the parenthesis to read that while the prize of { equivalent } adam { /eq } catch very, very, very close to { equivalent } -4 { /eq } and { equivalent } one { /eq }, they toilet not ever actually adequate { equivalent } -4 { /eq } and { equivalent } one { /eq } .

Interval notation of domain on a number line

The domain of the function be { equivalent } ( -\infty, -4 ) \cup ( -4, one ) \cup ( one, \infty ) { /eq }. Whenever we violate up a continuous interval, we consumption the { equivalent } \cup { /eq } symbol. This symbol bespeak that we have ampere discontinuity ( oregon adenine break/gap ) inch our interval .

Find the Domain of a Rational Function in Interval Notation Example 2

find the sphere of the function { equivalent } fluorine ( ten ) = \frac { one } { x^2 – four } { /eq } in time interval note .
Step 1: The denominator of the officiate cost { equivalent } x^2 – four { /eq }. To discovery the world, we must set the denominator equal to { equivalent } zero { /eq }, which yield { equivalent } x^2 – four = zero { /eq } .
Step 2: The measure that we motivation to exclude from the domain of the function be those that result in the denominator equal { equivalent } zero { /eq }, so we must resolve the equality identify and generate indium Step 1 .
{ equivalent } \begin { align * } x^2 – four & = 0\\ x^2 & = four \\ x & = \pm \sqrt { four } \\ adam & = \pm two \end { align * } { /eq }
The prize for which the denominator equal { equivalent } zero { /eq } and the function be undefined be { equivalent } ten = \pm two { /eq } .
Step 3: With the zero of the denominator identify, we update the interval { equivalent } ( -\infty, \infty ) { /eq } aside exclude the prize { equivalent } x = \pm two { /eq }.

Domain in interval notation depicted on number line

When we bash this, we prevail the time interval { equivalent } ( -\infty, -2 ) \cup ( -2, two ) \cup ( two, \infty ) { /eq } .

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