Video Transcript
Simplify the function 𝑓 of 𝑥
is equal to 𝑥 squared minus 81 over 𝑥 cubed plus 729 and find its domain.
In this question, we will begin
by finding the domain of the function and then simplify it by factoring the
numerator and denominator and canceling any shared factors. We recall that the domain of a
rational function is all real values of 𝑥 except where the denominator is equal
to zero. We therefore need to set the
cubic polynomial 𝑥 cubed plus 729 equal to zero. In order to solve this
equation, we will begin by factoring the left-hand side. 729 is a cube number. It is equal to nine cubed. This means that 𝑥 cubed plus
729 is written in the form 𝑥 cubed plus 𝑎 cubed. The sum of cubes formula states
that this is equal to 𝑥 plus 𝑎 multiplied by 𝑥 squared minus 𝑎𝑥 plus 𝑎
squared. 𝑥 cubed plus 729 is therefore
equal to 𝑥 plus nine multiplied by 𝑥 squared minus nine 𝑥 plus 81.
At this stage, we have the
product of a linear and quadratic term equal to zero. Setting the linear factor equal
to zero gives us 𝑥 is equal to negative nine. The quadratic equation 𝑥
squared minus nine 𝑥 plus 81 equals zero has no real solutions. This is because the
discriminant 𝑏 squared minus four 𝑎𝑐 is less than zero. We therefore have only one real
solution to the cubic equation 𝑥 cubed plus 729 equals zero. It is 𝑥 is equal to negative
nine. As this is the only value of 𝑥
that makes the function undefined, we can conclude that the domain of 𝑓 of 𝑥
is the set of all real values minus the set containing negative nine.
Let’s now move on to
simplifying the function. The numerator is written in the
form 𝑥 squared minus 𝑎 squared and can therefore be factored using the
difference of two squares. This is equal to 𝑥 plus 𝑎
multiplied by 𝑥 minus 𝑎. Since the square root of 81 is
nine, 𝑓 of 𝑥 is equal to 𝑥 plus nine multiplied by 𝑥 minus nine over 𝑥 plus
nine multiplied by 𝑥 squared minus nine 𝑥 plus 81. Since 𝑥 cannot equal negative
nine, we can cancel the shared factor of 𝑥 plus nine from the numerator and
denominator. 𝑓 of 𝑥 simplifies to 𝑥 minus
nine over 𝑥 squared minus nine 𝑥 plus 81. And we now have answers to the
two parts of the question.
