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For the following directed graph, let E = {I,J,K,L, M} and let V = {a,b, c, d}. d J M K For each definition of f below, determine if f is a function. If it is a function, state its domain and codomain in the form “f:X → Y" and give a table of values that lists each element x of the domain along with the corresponding element f (x) of the codomain. If the given f is not a function, explain why not. a. If x is an edge, then f (x) is the vertex that the edge points to. 5.

For the following directed graph, let E = {I,J,K,L, M} and let V = {a,b, c, d}. d J M K For each definition of f below, determine if f is a function. If it is a function, state its domain and codomain in the form “f:X → Y" and give a table of values that lists each element x of the domain along with the corresponding element f (x) of the codomain. If the given f is not a function, explain why not. a. If x is an edge, then f (x) is the vertex that the edge points to. 5.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PLEASE ANSWER ALL PARTS OF THIS QUESTION.

For the following directed graph, let ? = {?,?,?, ?, ?} and let ? = {?, ?, ?, ?}.

For the definition of ? below, determine if ? is a function. If it is a function, state its domain and codomain in the form “?:? → ?” and give a table of values that lists each element ? of the domain along with the corresponding element ?(?) of the codomain. If the given ? is not a function, explain why not.

If ? is an edge, then ?(?) is the vertex that the edge points to.

**Directed Graph Analysis** **Graph Description:** The graph is a directed graph with vertices \( V = \{a, b, c, d\} \) and edges \( E = \{I, J, K, L, M\} \). The directed edges are as follows: - \( I \) connects vertex \( a \) to vertex \( d \). - \( J \) connects vertex \( a \) to vertex \( b \). - \( K \) connects vertex \( b \) to vertex \( c \). - \( L \) connects vertex \( d \) to vertex \( c \). - \( M \) connects vertex \( d \) to vertex \( b \). **Task:** For each definition of \( f \) below, determine if \( f \) is a function. If it is a function, state its domain and codomain in the form " \( f: X \to Y \) " and provide a table of values that lists each element \( x \) of the domain along with the corresponding element \( f(x) \) of the codomain. If the given \( f \) is not a function, explain why not. **Definition:** a. If \( x \) is an edge, then \( f(x) \) is the vertex that the edge points to. **Analysis:** For each edge: - \( f(I) = d \) - \( f(J) = b \) - \( f(K) = c \) - \( f(L) = c \) - \( f(M) = b \) The function \( f \) is a well-defined function since each edge in \( E \) maps to exactly one vertex in \( V \). **Domain and Codomain:** - **Domain**: \( E = \{I, J, K, L, M\} \) - **Codomain**: \( V = \{a, b, c, d\} \) - **Function Notation**: \( f: E \to V \) **Table of Values:** | \( x \) | \( f(x) \) | |---------|------------| | I | d | | J | b | | K | c | | L | c | | M | b |
Transcribed Image Text:**Directed Graph Analysis** **Graph Description:** The graph is a directed graph with vertices \( V = \{a, b, c, d\} \) and edges \( E = \{I, J, K, L, M\} \). The directed edges are as follows: - \( I \) connects vertex \( a \) to vertex \( d \). - \( J \) connects vertex \( a \) to vertex \( b \). - \( K \) connects vertex \( b \) to vertex \( c \). - \( L \) connects vertex \( d \) to vertex \( c \). - \( M \) connects vertex \( d \) to vertex \( b \). **Task:** For each definition of \( f \) below, determine if \( f \) is a function. If it is a function, state its domain and codomain in the form " \( f: X \to Y \) " and provide a table of values that lists each element \( x \) of the domain along with the corresponding element \( f(x) \) of the codomain. If the given \( f \) is not a function, explain why not. **Definition:** a. If \( x \) is an edge, then \( f(x) \) is the vertex that the edge points to. **Analysis:** For each edge: - \( f(I) = d \) - \( f(J) = b \) - \( f(K) = c \) - \( f(L) = c \) - \( f(M) = b \) The function \( f \) is a well-defined function since each edge in \( E \) maps to exactly one vertex in \( V \). **Domain and Codomain:** - **Domain**: \( E = \{I, J, K, L, M\} \) - **Codomain**: \( V = \{a, b, c, d\} \) - **Function Notation**: \( f: E \to V \) **Table of Values:** | \( x \) | \( f(x) \) | |---------|------------| | I | d | | J | b | | K | c | | L | c | | M | b |
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