Identifying functions from mapping diagrams Grade 8 tutorial

📚 Understanding Functions from Mapping Diagrams

A mapping diagram is a visual way to represent a relation between two sets. It helps us easily see how elements from one set (the input or domain) are related to elements in another set (the output or range). A relation is a function if each input has exactly one output.

🕰️ History and Background

The concept of a function has evolved over centuries. Early notions were implicit in ancient mathematical writings, but the formal definition emerged in the 17th century with mathematicians like Leibniz and Bernoulli. Visual representations like mapping diagrams became popular later to aid understanding, especially in introductory algebra.

🔑 Key Principles

• 🌍 Domain and Range: The domain is the set of all possible inputs, and the range is the set of all possible outputs. In a mapping diagram, the domain is usually on the left, and the range is on the right.
• ➡️ Mapping: Arrows connect elements from the domain to their corresponding elements in the range.
• 💡 Function Definition: A relation is a function if each element in the domain maps to exactly one element in the range. No input can have multiple arrows coming from it.
• 🚫 Non-Function: If any element in the domain maps to more than one element in the range, the relation is not a function.

📝 How to Identify a Function

1. 🔍 Examine the Domain: Look at each element in the domain (the set on the left).
2. ➡️ Check the Arrows: For each element in the domain, see how many arrows are coming out of it.
3. ✅ Function: If each element in the domain has only one arrow coming out of it, the mapping diagram represents a function.
4. ❌ Not a Function: If any element in the domain has more than one arrow coming out of it, the mapping diagram does not represent a function.

🧪 Real-World Examples

Let's look at some examples to clarify how to identify functions from mapping diagrams.

Example 1: Function

Consider a mapping diagram where the domain is $A = \{1, 2, 3\}$ and the range is $B = \{4, 5, 6\}$. The mapping is as follows:

• 1 maps to 4
• 2 maps to 5
• 3 maps to 6

Since each element in $A$ maps to exactly one element in $B$, this is a function.

Example 2: Not a Function

Consider a mapping diagram where the domain is $A = \{1, 2, 3\}$ and the range is $B = \{4, 5, 6\}$. The mapping is as follows:

• 1 maps to 4
• 2 maps to 5
• 3 maps to both 5 and 6

Since the element 3 in $A$ maps to two elements (5 and 6) in $B$, this is not a function.

Example 3: Function with Repeated Range Values

Consider a mapping diagram where the domain is $A = \{1, 2, 3\}$ and the range is $B = \{4, 5, 6\}$. The mapping is as follows:

• 1 maps to 4
• 2 maps to 4
• 3 maps to 5

Even though 1 and 2 both map to 4, each element in $A$ still maps to only one element in $B$. Therefore, this is a function.

📝 Practice Quiz

Determine whether each of the following mapping diagrams represents a function:

1. Domain: $\{a, b, c\}$, Range: $\{x, y, z\}$. Mappings: $a \to x$, $b \to y$, $c \to z$.
2. Domain: $\{p, q, r\}$, Range: $\{u, v\}$. Mappings: $p \to u$, $q \to u$, $r \to v$.
3. Domain: $\{1, 2, 3\}$, Range: $\{a, b\}$. Mappings: $1 \to a$, $2 \to a$, $3 \to a$.
4. Domain: $\{4, 5, 6\}$, Range: $\{x, y, z\}$. Mappings: $4 \to x$, $5 \to y$, $6 \to x$.
5. Domain: $\{a, b, c\}$, Range: $\{1, 2, 3\}$. Mappings: $a \to 1$, $b \to 2$, $c \to 1$.
6. Domain: $\{1, 2\}$, Range: $\{x, y, z\}$. Mappings: $1 \to x$, $2 \to y$, $2 \to z$.
7. Domain: $\{p, q, r\}$, Range: $\{a, b\}$. Mappings: $p \to a$, $q \to b$, $r \to a$.

Answers:

1. Function
2. Function
3. Function
4. Function
5. Function
6. Not a Function
7. Function

🔑 Conclusion

Identifying functions from mapping diagrams is a fundamental skill in understanding relations and functions. By ensuring that each input has exactly one output, you can quickly determine whether a given mapping diagram represents a function. This concept is crucial for further studies in algebra and beyond.