### Video Transcript

Identify the domain of π of π₯ is
equal to nine π₯ plus eight all divided by three π₯ plus two.

In this question, weβre given a
function π of π₯, which we can see is the quotient of two linear functions. In other words, this is a rational
function. And weβre asked to identify the
domain of this function, and thereβs a few different ways of doing this. Letβs start by recalling exactly
what we mean by the domain of a function.

We recall that when we say the
domain of a function, what we mean is the set of inputs for that function. So in this question, when weβre
asked to find the domain of our function π of π₯, we want to find all of the
possible inputs for this function. Now we could do this formally by
looking for all of the values where our function π of π₯ is not defined and the
values of π₯ where it is defined. However, weβve already done this in
general for all rational functions, so we can actually make this a lot simpler.

Weβve actually shown that for a
rational function to not be defined, the denominator has to be equal to zero. In other words, the domain of any
rational function is all real values except the zeros of the denominator. Therefore, instead of finding the
domain of our function π of π₯ by using the definition of a domain, all we need to
do is solve the denominator is equal to zero. Setting the denominator equal to
zero, we get three π₯ plus two is equal to zero. And of course, we can solve this
equation for π₯. We subtract two from both sides of
the equation, giving us three π₯ minus two, and then we divide through by three. This gives us that π₯ is equal to
negative two over three.

Therefore, our function π of π₯ is
not defined when π₯ is equal to negative two over three since weβre dividing by
zero. And for all other input values of
π₯, weβll be taking the quotient of two real numbers where the denominator is not
equal to zero. So our function is defined for
these values of π₯. So weβve shown the domain of our
function π of π₯ is all real numbers except for negative two over three. And of course, we can represent
this in set notation as the set of all real numbers excluding negative two over
three.

And itβs important we do this
because, remember, the domain of a function is a set. Therefore, we were able to find the
domain of the function π of π₯ is equal to nine π₯ plus eight all divided by three
π₯ plus two by finding the zeros of the denominator. We were able to show the domain of
this function is the set of all real numbers excluding negative two over three.