Question Video: Finding the Domain of Composite Functions | Nagwa Question Video: Finding the Domain of Composite Functions | Nagwa

Question Video: Finding the Domain of Composite Functions Mathematics

If the function 𝑓(𝑥) = 2/𝑥, where 𝑥 ≠ 0, and the function 𝑔(𝑥) = 𝑥 − 41, determine the domain of 𝑓 ∘ 𝑔.

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Video Transcript

If the function 𝑓 of 𝑥 equals two over 𝑥, where 𝑥 is not equal to zero, and the function 𝑔 of 𝑥 equals 𝑥 minus 41, determine the domain of 𝑓 of 𝑔.

We recall first that this notation of 𝑓 and then a little circle and then 𝑔 means 𝑓 of 𝑔. It’s the composite function 𝑓 of 𝑔 of 𝑥. This is the composite function we get when we take an input, apply the function 𝑔, and then apply the function 𝑓 to the result. We’re given the definitions of the two functions 𝑓 and 𝑔 in the question and asked to determine the domain of this composite function.

Now we recall that the domain of a function is the set of all input values to that function. Or we can think of it as the set of all values on which the function acts. We should notice that the domain of the function 𝑓 of 𝑥 is actually given in the question; it’s 𝑥 not equal to zero. And this is because in the function 𝑓, we are dividing two by 𝑥. And because division by zero is undefined, the function 𝑓 can’t take the value zero as an input, but it can take any other real value.

Remember though we want to find the domain of the composite function 𝑓 of 𝑔 of 𝑥. So, let’s begin by finding an algebraic expression for 𝑓 of 𝑔 of 𝑥. If we start with an input 𝑥 and then apply the function 𝑔 first of all, we have 𝑔 of 𝑥 is equal to 𝑥 minus 41. We then need to apply the function 𝑓, which means we take 𝑔 of 𝑥 — that’s 𝑥 minus 41 — as our input to the function 𝑓. So, we have 𝑓 of 𝑥 minus 41. 𝑓 is the function that divides two by its input value. So, 𝑓 of 𝑥 minus 41 is two over 𝑥 minus 41.

We have then an algebraic expression for the composite function 𝑓 of 𝑔 of 𝑥, and we can now consider its domain. This expression will be well defined, provided the denominator is not equal to zero. In other words, it will be undefined if the denominator is equal to zero, so if 𝑥 minus 41 is equal to zero, which leads to 𝑥 equals 41. This means that the function can’t act on the value 41 because it would lead to two divided by zero, which is undefined. The function can act on all other real values of 𝑥.

So, the domain of the composite function 𝑓 of 𝑔 is the set of all real numbers minus the value 41.

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