Question Video: Simplifying a Rational Function and Identifying Its Domain | Nagwa Question Video: Simplifying a Rational Function and Identifying Its Domain | Nagwa

Question Video: Simplifying a Rational Function and Identifying Its Domain Mathematics • Third Year of Preparatory School

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Simplify the function π(π₯) = (2/(π₯ + 2)) Γ ((π₯Β² + 6π₯ + 8)/2π₯), and determine its domain.

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Video Transcript

Simplify the function π of π₯ equals two over π₯ plus two times π₯ squared plus six π₯ plus eight over two π₯, and determine its domain.

It generally makes sense to determine the domain of a rational function before simplifying it. By doing so, we can avoid losing any potential values of π₯ that could cause us issues later down the line. But before we can simplify π of π₯, letβs multiply the two fractions together. When we multiply pairs of fractions, we multiply their numerators and separately their denominators. So, π of π₯ is two times the quadratic π₯ squared plus six π₯ plus eight all over two π₯ times π₯ plus two.

So, before we simplify it, letβs determine the domain of the function. The domain is said to be the set of possible inputs to the function, in other words, the possible π₯-values that ensure π of π₯ is well defined. And since weβre dealing with a rational function, we know that the denominator of that function cannot be equal to zero. Otherwise, itβs simply the quotient of two polynomial functions. And the domain of a polynomial function is the set of real numbers. So, the domain of π of π₯ will be the set of real numbers. But we will need to exclude any values of π₯ that make the denominator equal to zero.

To find those values of π₯, weβll set the denominator equal to zero and solve for π₯. In other words, two π₯ times π₯ plus two equals zero. And of course, since we are finding the product of two π₯ and π₯ plus two and getting an answer of zero, that will only be true if either of those expressions itself is equal to zero, in other words, if two π₯ equals zero or π₯ plus two equals zero. Well, if two π₯ equals zero, then π₯ itself must be zero. Similarly, if we subtract two from both sides of this equation, we get π₯ equals negative two. So, the domain is the set of real numbers not including the set containing the elements zero and negative two.

With that in mind, weβre now ready to simplify the expression. And when we simplify a function, weβre looking to find common factors. We begin by noticing that there is a shared factor of two on the numerator and denominator of our fraction. We therefore divide through by two, leaving us with π₯ squared plus six π₯ plus eight over π₯ times π₯ plus two. Then, at this stage, weβve actually run out of shared factors, so we look to factor the numerator. π₯ squared plus six π₯ plus eight can in fact be written as π₯ plus four times π₯ plus two.

Then we notice that the fraction π₯ plus four π₯ plus two over π₯ times π₯ plus two has a common factor of π₯ plus two on the numerator and the denominator. Since π₯ cannot be equal to negative two, weβre therefore able to divide the numerator and denominator by π₯ plus two. And that leaves us simply with π₯ plus four over π₯. So, the function π of π₯ simplifies to π₯ plus four over π₯. And its domain is the set of real numbers minus the set containing the elements zero and negative two.

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