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:
2a. F + g
2b. F - g
2c. F . g
2d. F/g
2e. G o f o h
To find the given expressions and state the domain, we need to perform the respective operations on the given functions and then determine the domain of the resulting function. Let's calculate each expression step by step:
2a. F + g:
To add two functions, we need to add their corresponding values.
F = (1, 2), (2, 3), (3, 4), (4, 5)
G = (1, -2), (3, -3), (5, -5)
Adding them term by term:
F + G = ((1, 2 + (-2)), (2, 3 + 0), (3, 4 + (-3)), (4, 5 + 0))
= ((1, 0), (2, 3), (3, 1), (4, 5))
The resulting function F + G is ((1, 0), (2, 3), (3, 1), (4, 5)).
Domain: The domain is the set of all x-values in the function. Therefore, the domain for F + G is {1, 2, 3, 4}.
2b. F - g:
Subtracting two functions works similarly to addition.
F - G = ((1, 2 - (-2)), (2, 3 - 0), (3, 4 - (-3)), (4, 5 - 0))
= ((1, 4), (2, 3), (3, 7), (4, 5))
The resulting function F - G is ((1, 4), (2, 3), (3, 7), (4, 5)).
Domain: The domain for F - G is {1, 2, 3, 4}.
2c. F . g:
To find the dot product of two functions, we need to multiply their corresponding values.
F = (1, 2), (2, 3), (3, 4), (4, 5)
G = (1, -2), (3, -3), (5, -5)
Multiplying them:
F . G = (1 * (-2) + 2 * (-3) + 3 * (-3) + 4 * (-5))
= (-2 - 6 - 9 - 20)
= -37
The dot product of F and G is -37.
Domain: Since the dot product is a single value, it doesn't have a domain. It exists independently of any specific x or y values.
2d. F / g:
To divide two functions, we need to divide their corresponding values.
F = (1, 2), (2, 3), (3, 4), (4, 5)
G = (1, -2), (3, -3), (5, -5)
Dividing them term by term:
F / G = ((1, 2 / (-2)), (2, 3 / (-3)), (3, 4 / (-5)), (4, 5 / (-5)))
= ((1, -1), (2, -1), (3, -0.8), (4, -1))
The resulting function F / G is ((1, -1), (2, -1), (3, -0.8), (4, -1)).
Domain: The domain for F / G is {1, 2, 3, 4}.
2e. G o f o h:
The notation G o f o h represents the composition of three functions, in the order of h, f, G. To find the composite function, we evaluate the functions in that order.
H = (1, 0), (2, 1), (3, 2)
F = (1, 2), (2, 3), (3, 4), (4, 5)
G = (1, -2), (3, -3), (5, -5)
Evaluating G at the output of f o h:
f o h = ((1,2), (2,3), (3,4), (4,5))
G o f o h = G(f o h) = G((1,2), (2,3), (3,4), (4,5))
Evaluating G at each point:
G(1,2) = -2
G(2,3) = undefined (No value for x=2 in G)
G(3,4) = -3
G(4,5) = undefined (No value for x=4 in G)
The resulting function G o f o h is (-2, -3).
Domain: The domain for G o f o h is {1, 3}.
I hope this explanation helps you understand how to perform these operations and determine the domain for each resulting function. Let me know if you have any further questions!