Video Transcript
Determine the domain of the
function 𝑓 of 𝑥 is equal to three divided by 𝑥 minus three plus one divided by 𝑥
plus four.
The question wants us to find the
domain of the function 𝑓 of 𝑥. We recall the domain of a function
is the set of possible inputs of our function. In our case, we can see that our
function 𝑓 of 𝑥 is defined to be the sum of two functions. In fact, it’s the sum of two
rational functions. Remember, we call a function
rational if it’s the quotient between two polynomials. A linear function and a constant
function are both examples of polynomials. So we need to find the set of
possible inputs for our function, which is the sum of two rational functions.
Let’s start by recalling what we
know about the domains of each of our rational functions individually. First, we recall any rational
function will be defined everywhere except where its denominator is equal to
zero. Remember, this is because we can
input any value of 𝑥, and we’ll get a real number divided by another real
number. But we can never divide by
zero. It will always be undefined.
So let’s look at each term of 𝑓 of
𝑥 individually. Let’s start with three divided by
𝑥 minus three. We see this is a rational
function. This will be defined everywhere
except where its denominator is equal to zero. And we know its denominator is
equal to zero only when 𝑥 is equal to three. We can do exactly the same with the
second term one divided by 𝑥 plus four. This is also a rational function,
so it will be defined everywhere except where its denominator is equal to zero,
which in this case is when 𝑥 is equal to negative four.
But let’s think about what this
means for our function 𝑓 of 𝑥. For example, if we were to
substitute 𝑥 is equal to three into 𝑓 of 𝑥, we’d get 𝑓 of three is equal to
three divided by three minus three plus one divided by three plus four. And if we simplify this, we get
three over zero plus one over seven. Of course, three divided by zero is
undefined. We can’t divide by zero. And of course, the same will be
true when 𝑥 is equal to negative four. Since we’ve already explained one
divided by 𝑥 plus four is not defined when 𝑥 is equal to negative four.
Every other input of 𝑥 will just
give us the sum of two real numbers. So we’ve shown the domain of our
function 𝑓 of 𝑥 is equal to three divided by 𝑥 minus three plus one divided by 𝑥
plus four is all real numbers except when 𝑥 is equal to negative four and when 𝑥
is equal to three.
