Question 1183975: If f = {(1, 2), (2, 3), (3, 4), (4, 5)}, g = {(1, -2), (3, -3), (5, -5)}, and h = {(1, 0), (2, 1), (3, 2)}, find the following and state the domain.
1. f + g
2. f - g
3. f * g
4. f / h
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
I'll go over problem 1.
Given functions:
f = {(1, 2), (2, 3), (3, 4), (4, 5)}
g = {(1, -2), (3, -3), (5, -5)}
h = {(1, 0), (2, 1), (3, 2)}
We only need to focus on functions f and g for this problem.
The f(x) function set tells us
f(1) = 2
f(2) = 3
f(3) = 4
f(4) = 5
In general the rule is f(x) = y where (x,y) is an element of the function f
note that f(5) is not defined
For g(x), we have
g(1) = -2
g(3) = -3
g(5) = -5
note that g(2) and g(4) are not defined.
Now onto computing f+g
Effectively what we do is for each input (x = 1 through x = 5), we determine the y output value by adding the outputs of f(x) and g(x)
Let's compute the output when x = 1 is the input
(f+g)(x) = f(x)+g(x)
(f+g)(1) = f(1)+g(1)
(f+g)(1) = 2+(-2)
(f+g)(1) = 0
When x = 1 is the input, y = 0 is the output for the f+g function.
This means the ordered pair (1,0) is part of the f+g function set.
Repeat for x = 2
(f+g)(x) = f(x)+g(x)
(f+g)(2) = f(2)+g(2)
(f+g)(2) = 3+undefined
(f+g)(2) = undefined
The g(2) is undefined as there isn't an (x,y) ordered pair with x = 2 in the g(x) function. This undefined property carries down to the final result. So there isn't any output for f+g when the input is 2.
Repeat for x = 3
(f+g)(x) = f(x)+g(x)
(f+g)(3) = f(3)+g(3)
(f+g)(3) = 4+(-3)
(f+g)(3) = 1
This time both f(x) and g(x) are defined for this input
The input x = 3 leads to the output y = 1 for the f+g function
So (3,1) is part of the f+g function set
This process is continued until we reach x = 5
You should have the following set:
f+g = {(1,0), (3,1)}
Earlier we found that f+g isn't defined when x = 2, so we ignore this input. Also, the x values x = 4 and x = 5 are not included because g(x) isn't defined when x = 4 and f(x) isn't defined when x = 5. Both functions need to be defined.
In short, we ignore any inputs that lead to undefined outputs.
The domain of f+g is simply the list of x coordinates of each (x,y) point. Therefore, the domain is {1,3}
The domain is the set of all allowed inputs of a function.
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Answers for problem 1
f+g = {(1,0), (3,1)}
Domain = {1,3}
I'll let you tackle problems 2 through 4. They follow the same basic idea. You need to go through each input (x = 1 through x = 5) and apply the arithmetic operation shown. If a function isn't defined at a particular input, then the whole thing isn't defined. Keep in mind that division by zero isn't defined either, so make sure that h(x) is nonzero as it's in the denominator.
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