Question

Given that f(x) = {(1, 3), (2, 9), (5, 7), (11, 9)} and g(x) = {(2, 8), (3, 1), (4, 5), (9, 1), (11, 0)}

Determine the following.
a. f + g(x)
b. g(x) − f(x)
c. f × g(x)
d. f(x) / g(x)
e. f∘ g(x)

Answer 1

Operations on the given functions f(x) and g(x) can be performed: addition and subtraction result in sums and differences of corresponding y-values; multiplication results in products; division must avoid division by zero; and composition cannot be determined with the provided sets.

Step-by-step explanation:

Given two functions f(x) and g(x) as sets of ordered pairs, we can perform various operations such as summation, subtraction, multiplication, and division, as long as we only consider x-values that appear in both functions.

1. For f + g(x), we sum the y-values of both functions for the common x-values. In this case, the common x-values are 2 and 11. Therefore, f + g(2) = f(2) + g(2) = 9 + 8 = 17, and f + g(11) = f(11) + g(11) = 9 + 0 = 9.
2. For g(x) - f(x), we subtract the y-values of f from g for the common x-values, resulting in g - f(2) = g(2) - f(2) = 8 - 9 = -1 and g - f(11) = g(11) - f(11) = 0 - 9 = -9.
3. In the case of f × g(x), we multiply the y-values for common x-values. Thus, f × g(2) = f(2) × g(2) = 9 × 8 = 72 and f × g(11) = f(11) × g(11) = 9 × 0 = 0.
4. For f(x) / g(x), we divide the y-values of f by those of g for common x-values, giving f / g(2) = f(2) / g(2) = 9 / 8 and f / g(11) = f(11) / g(11) = 9 / 0. However, since division by zero is undefined, f(x) / g(x) is not defined for x = 11.
5. The composition of functions, represented as f∘g(x), means we find g(x) first and then apply the result as an input to f(x). Unfortunately, for the given sets, there are no x-values in g(x) that correspond to x-values in f(x), so the composition f∘g(x) cannot be determined from the provided ordered pairs.

Answer 2

The composite functions are (f + g)(x) = {(2, 17), (11, 9)}, g(x) - f(x) = {(2, -1), (11, -9)}, (f * g)(x) = {(2, 72), (11, 0)}, (f / g)(x) = {(2, 9/8)} and f o g(x) = undefined

How to evaluate the composite functions

From the question, we have the following parameters that can be used in our computation:

f(x) = {(1, 3), (2, 9), (5, 7), (11, 9)}

Also, we have

g(x) = {(2, 8), (3, 1), (4, 5), (9, 1), (11, 0)}

Also, we only use the function values where they have common inputs

The composite functions are then calculated as

(f + g)(x) = f(x) + g(x)

This gives

(f + g)(x) = {(2, 9 + 8), (11, 9 + 0)}

(f + g)(x) = {(2, 17), (11, 9)}

g(x) - f(x) = {(2, 8 - 9), (11, 0 - 9)}

g(x) - f(x) = {(2, -1), (11, -9)}

(f * g)(x) = f(x) * g(x)

This gives

(f * g)(x) = {(2, 9 * 8), (11, 9 * 0)}

(f * g)(x) = {(2, 72), (11, 0)}

(f / g)(x) = f(x) / g(x)

This gives

(f / g)(x) = {(2, 9/8), (11, 9/0)}

(f / g)(x) = {(2, 9/8)}

f o g(x) = f(g(x))

So, we have

f o g(x) = {f(g(1)), f(g(2)), f(g(5)), f(g(11))}

f o g(x) = {f(8), f(0)}

Evaluate

f o g(x) = undefined

Hence, the composite functions are (f + g)(x) = {(2, 17), (11, 9)}, g(x) - f(x) = {(2, -1), (11, -9)}, (f * g)(x) = {(2, 72), (11, 0)}, (f / g)(x) = {(2, 9/8)} and f o g(x) = undefined