Video: Finding the Domains of Composite Functions

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.