Question 1194815: Let 𝑓 = {(3, 8), (2, 5), (4, −5), (9, 3 )} and 𝑔 = {(9, 2), (−5, 3), (5, 9 ), (8; 10), (1; 9)}. Determine the
compositions (𝑓 ∘ 𝑔) and (𝑔 ∘ 𝑓) if they exist.
1) (𝑓 ∘ 𝑔)
2) (𝑔 ∘ 𝑓)
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
I'll do question number 1 to get you started.
The notation (𝑓 ∘ 𝑔)(x) is the same as f(g(x))
We have g(x) as the inner function, which means it goes first in the computation order.
The first x coordinate mentioned in g(x) is x = 9
It's part of the ordered pair (9,2)
Since the input x = 9 leads to the output y = g(x) = 2, this means
f(g(9)) = f(2)
Then we look through f(x) to see if there's an ordered pair with x = 2
There is such a point and it is (2,5)
f(g(9)) = f(2) = 5
f(g(9)) = 5
(𝑓 ∘ 𝑔)(9) = 5
Therefore, (9,5) is one ordered pair in the function (𝑓 ∘ 𝑔)(x)
The next point mentioned in g(x) is (-5,3)
x = -5 leads to y = g(x) = 3
f(g(-5)) = f(3) = 8
(𝑓 ∘ 𝑔)(-5) = 8
So (-5,8) is another point in the final answer.
Repeat this process for the remaining other x coordinates of the points in g(x). I'll let you do these steps.
Take note that
x = 8 leads to g(x) = 10
f(g(8)) = f(10) = undefined
because there isn't a point in f(x) with an x coordinate of 10.
Therefore, x = 8 won't be mentioned in the final answer. We would say that x = 8 is not in the domain of 𝑓 ∘ 𝑔
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Final answer for question 1:
{ (9,5), (-5, 8), (5, 3), (1, 3) }
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