Video Transcript
If the function 𝑓 of 𝑥 equals 17
over 𝑥, where 𝑥 is not equal to zero, and the function 𝑔 of 𝑥 equals 𝑥 squared
minus 361, determine the domain of 𝑓 of 𝑔 of 𝑥.
We recall, firstly, that this
notation, 𝑓 and then a small circle and then 𝑔 of 𝑥, means the composite function
found when we apply the function 𝑔 first and then apply the function 𝑓 to the
result. We’re asked to determine the domain
of this composite function. And we recall then that the domain
of a function is the set of all values on which the function acts. Notice that, in the definition of
the function 𝑓 of 𝑥, we’re told that this is valid for 𝑥 not equal to zero. And this is because if we were to
try to evaluate 𝑓 of zero, we would have 17 over zero, which is undefined. So, the domain of the function 𝑓
of 𝑥 is all real numbers except zero.
For the function 𝑔 of 𝑥, no
restrictions have been given because there are no values that cause problems. We can square any value and then we
can subtract 361 without running into problems with values being undefined. We’ll consider two approaches to
this question. The first is to find an algebraic
expression for the composite function 𝑓 of 𝑔 of 𝑥. So, we start with an 𝑥-value,
apply the function 𝑔 of 𝑥, which will give 𝑥 squared minus 361, and then take
this as our input to the function 𝑓. 𝑓 of 𝑥 is the function where we
divide 17 by our input. So, 𝑓 of 𝑥 squared minus 361 is
17 over 𝑥 squared minus 361. And so, we have a general
expression for the composite function 𝑓 of 𝑔 of 𝑥.
Now, notice that this will be
undefined when the denominator of the fraction is equal to zero. That’s when 𝑥 squared minus 361 is
equal to zero. We can solve this equation by
adding 361 to each side and then taking the square root. The square root of 361 is 19, so we
find that 𝑓 of 𝑔 of 𝑥 will be undefined when 𝑥 equals plus or minus 19. This means our function 𝑓 of 𝑔 of
𝑥 can act on all values except positive or negative 19, so we can express its
domain in two ways. We can either write 𝑥 is not equal
to positive or negative 19. Or we can write the domain as the
set of all real numbers minus the set containing the two elements negative 19 and
positive 19. Either of these two styles of
notation would be absolutely fine.
Now, another way to approach this
problem would be to use the fact that our second function 𝑓 of 𝑥 can’t act on the
value zero. If 𝑓 of zero is undefined, then
this means that the composite function 𝑓 of 𝑔 of 𝑥 will be undefined when 𝑔 of
𝑥 is equal to zero. That is, when the first function
gives the value of zero, which we then attempt to input into our second
function. 𝑔 of 𝑥 is the function 𝑥 squared
minus 361. So, to find where 𝑔 of 𝑥 is equal
to zero, we’d be solving this quadratic equation. But this just brings us to the same
stage of working out as we had here in our previous method. So, either of these approaches
would be absolutely fine. And in both cases, we conclude that
the domain of the composite function 𝑓 of 𝑔 of 𝑥 is all of the real numbers apart
from negative 19 and positive 19.
