Is my answer correct to this question? QUESTION:

1. There are 5 students; Ani, Leon, Linh, Liam, and Abdul with scores in computer science as 75, 60, 85, 95, and 60 respectively. They are graded by the University as B, C, B+, A, and C respectively. Define the mappings from students to marks and marks to grades explicitly.

• What are the domains and ranges of the functions?
• Can you make a composite function out of this? If yes, make the composite function and find if the composition of functions is commutative. Explain the reason.

THIS IS MY ANSWER:

Mapping from students to marks:

Ani -> 75

Leon -> 60

Linh -> 85

Liam -> 95

Abdul -> 60

Mapping from marks to grades:

75 -> B

60 -> C

85 -> B+

95 -> A

The domain of the function from students to marks is the set of students: {Ani, Leon, Linh, Liam, Abdul}.

The range of the function from students to marks is the set of marks: {75, 60, 85, 95}.

The domain of the function from marks to grades is the set of marks: {75, 60, 85, 95}.

The range of the function from marks to grades is the set of grades: {B, C, B+, A}.

A composite function is made by combining the mapping from students to marks and the mapping from marks to grades. The composite function will map each student directly to their corresponding grade.

Composite function: "Grades" composed of "Marks" composed of "Students"

Grades(Students(x)) = Marks(x)

For example, Grades(Ani) = Marks(Ani) = 75, which corresponds to the grade B.

To check if the composition of functions is commutative, we need to verify if the order of composition affects the result. In this case, if we first apply the function "Students" and then the function "Marks", or if we first apply the function "Marks" and then the function "Students", do we get the same result?

In our case, applying "Students" and then "Marks" means finding the grade for a given student's mark. Applying "Marks" and then "Students" means finding the student for a given grade.

Since there is a one-to-one correspondence between marks and grades, and each mark is uniquely associated with a student, and vice versa, the composition of functions is commutative. The order of applying the functions does not change the final result.

9:02 AM Fri 30 Jun . . VPN 99% discretemath.org is used to denote that element of B to which a is related. f(a) is called the 138 of 577 image of a, or, more precisely, the image of a under f. We write f(a) = b to indicate that the image of a is b. In Example 7.1.5, the image of 2 under f is 4; that is, f(2) = 4. In Example 7.1.2, the image of -1 under s is 1; that is, s(-1) = 1. Definition 7.1.7 Range of a Function. The range of a function is the set of images of its domain. If f : X - Y, then the range of f is denoted f(X), and f (X) = {f(a) |ac X} = {be Y | Fa E X such that f(a) = b}. D Note that the range of a function is a subset of its codomain. f(X) is also read as "the image of the set X under the function f" or simply "the image of In Example 7.1.2, s(A) = {0, 1, 4}. Notice that 2 and 3 are not images of any element of A. In addition, note that both 1 and 4 are related to more than one element of the domain: s(1) = s(-1) = 1 and s(2) = s(-2) = 4. This does not violate the definition of a function. Go back and read the definition if this isn't clear to you. In Example 7.1.3, the range of L is equal to its codomain, R. If b is any real number, we can demonstrate that it belongs to L(R) by finding a real number x for which L(x) = b. By the definition of L, L(x) = 3x, which leads us to the equation 3x = b. This equation always has a solution, "; thus L(R) = R. The formula that we used to describe the image of a real number under L, L(x) = 3x, is preferred over the set notation for L due to its brevity. Any time a function can be described with a rule or formula, we will use this form of description. In Example 7.1.2, the image of each element of A is its square