Focal Area 1: Relations and Functions Define the term “relation” and explain how it differs from a function…

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Focal Area 1: Relations and Functions Define the term “relation” and explain how it differs from a function. Use an example to illustrate your answer. Classify each of the following as either a relation or a function. Justify your answer for each: - A set of ordered pairs where each input has exactly one output - A set of ordered pairs where one input has multiple outputs Draw an arrow diagram to represent the following relation: Domain = 1, 2, 3, Co-domain = 2, 4, 6, 8, where each element in the domain is mapped to its double. Identify the type of relation shown in the following arrow diagram and explain why it belongs to that category: - Element A maps to 5 - Element B maps to 5 - Element C maps to 7 Given the relation R = (1, 2), (2, 4), (3, 6), (4, 8), identify the domain, co-domain, and range. Explain the difference between co-domain and range. Determine whether the following relation is a function: (2, 3), (3, 5), (4, 7), (5, 9). Explain your reasoning. Create a mapping diagram for the relation where each student is mapped to their favourite subject. Classify this relation as one-to-one, one-to-many, many-to-one, or many-to-many. Explain what a “many-to-one” relation is and provide a real-world example from your school context. Given two sets: Set A = red, blue, green and Set B = apple, sky, grass, create a one-to-one relation between them and represent it as ordered pairs. Identify which of the following mappings represent functions: - Mapping 1: Each person to their date of birth - Mapping 2: Each date of birth to a person Explain your answer for each. Focal Area 2: Rules of Mappings For the linear mapping shown below, determine the rule in the form y = mx + c: - When x = 0, y = 3 - When x = 1, y = 5 - When x = 2, y = 7 Given the mapping: x = 1, 2, 3, 4 and y = 5, 7, 9, 11, find the rule of the mapping and verify it with at least two pairs. Explain the difference between a linear mapping and an exponential mapping. Provide one example of each. For the following set of ordered pairs, determine if the mapping is linear or exponential: - (1, 2), (2, 4), (3, 8), (4, 16) Justify your answer. Given a linear mapping with the rule y = 3x + 2, calculate the output values for x = 0, 1, 2, and 3. Organize your results in a table. A mobile phone plan charges a fixed fee of GH₵10 plus GH₵0.50 per minute. Write the rule for this mapping in the form y = mx + c, where x represents minutes and y represents total cost. Identify the constant difference in the domain and co-domain for the following mapping, then determine the rule: - Domain: 2, 4, 6, 8 - Co-domain: 7, 13, 19, 25 For the mapping y = 2x + 1, create a table of values for x = −2, −1, 0, 1, 2 and explain what the values of m and c represent. Given the rule y = −2x + 5, determine the output when x = 3 and explain what this ordered pair represents in the context of the mapping. A taxi charges GH₵2.50 per kilometre plus a fixed charge of GH₵5. Write the mapping rule and calculate the cost for a 15 km journey.

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Focal Area 1: Relations and Functions

A relation is a set of ordered pairs that shows a relationship between two sets of values. A function is a special type of relation where each input (element from the domain) has exactly one output (element from the range). The key difference is that in a function, no input can be mapped to more than one output. For example, the relation $R_1 = \{(1, \text{apple}), (1, \text{banana}), (2, \text{orange})\}$ is a relation but not a function because the input 1 is mapped to two different outputs (apple and banana). The relation $R_2 = \{(1, \text{apple}), (2, \text{banana}), (3, \text{orange})\}$ is a function because each input has exactly one output.
• A set of ordered pairs where each input has exactly one output: This is a function. It satisfies the definition of a function, where every element in the domain corresponds to precisely one element in the co-domain.
- A set of ordered pairs where one input has multiple outputs: This is a relation. It does not satisfy the definition of a function because at least one input is mapped to more than one output.
Here is the arrow diagram:
$\begin{array}{c} \text{Domain} \\ \{1, 2, 3\} \end{array} \xrightarrow{\text{maps to its double}} \begin{array}{c} \text{Co-domain} \\ \{2, 4, 6, 8\} \end{array}$
- $1 \to 2$
- $2 \to 4$
- $3 \to 6$
This is a many-to-one relation. It belongs to this category because multiple distinct elements from the domain (Element A and Element B) are mapped to the same element in the co-domain (5).
Given the relation $R = \{(1, 2), (2, 4), (3, 6), (4, 8)\}$ :
- The domain is the set of all first elements: $\boxed{\{1, 2, 3, 4\}}$ .
- The co-domain is the set of all possible output values. Without further information, we can assume the co-domain is the set of all second elements present in the relation, or a larger set from which these values are drawn. If the question implies the co-domain is just the set of values that could be outputs, and the range is the actual outputs, then the co-domain could be a larger set. However, based on the given relation, the most direct interpretation for the co-domain is the set of all second elements: $\boxed{\{2, 4, 6, 8\}}$ .
- The range is the set of all second elements that are actually mapped to: $\boxed{\{2, 4, 6, 8\}}$ . The co-domain is the set of all possible output values for a function or relation, while the range is the set of all actual output values that are produced by the function or relation for the given domain. The range is always a subset of the co-domain. In this specific example, the range is equal to the co-domain.
The relation $\{(2, 3), (3, 5), (4, 7), (5, 9)\}$ is a function. Each input value (2, 3, 4, 5) is mapped to exactly one output value (3, 5, 7, 9 respectively). There are no input values that have more than one corresponding output value.
Here is an example mapping diagram for students to their favourite subject:
$\begin{array}{c} \text{Students} \\ \text{Alice} \\ \text{Bob} \\ \text{Charlie} \\ \text{David} \end{array} \xrightarrow{\text{favourite subject}} \begin{array}{c} \text{Subjects} \\ \text{Math} \\ \text{Science} \\ \text{English} \end{array}$
- Alice $\to$ Math
- Bob $\to$ Science
- Charlie $\to$ Math
- David $\to$ English This relation is a many-to-one relation because multiple students (Alice and Charlie) have the same favourite subject (Math).
A many-to-one relation is a type of relation where two or more distinct input values (from the domain) are mapped to the same output value (in the co-domain). A real-world example from a school context is: Students to their homeroom teacher. Multiple students (inputs) are assigned to the same homeroom teacher (output). For instance, all students in Class 7A have Mrs. Smith as their homeroom teacher.
Given Set A = {red, blue, green} and Set B = {apple, sky, grass}, a one-to-one relation can be created by pairing each element from Set A with a unique element from Set B. One possible one-to-one relation as ordered pairs is: $\boxed{\{(\text{red, apple}), (\text{blue, sky}), (\text{green, grass})\}}$
• Mapping 1: Each person to their date of birth. This mapping represents a function. Each person has exactly one date of birth, so every input (person) corresponds to precisely one output (date of birth).
- Mapping 2: Each date of birth to a person. This mapping does not represent a function. A single date of birth can correspond to multiple people (e.g., twins, or simply different individuals born on the same day). Therefore, one input (date of birth) can have multiple outputs (people).

Focal Area 2: Rules of Mappings

For a linear mapping $y = mx + c$ :

Use the point $(0, 3)$ to find $c$ .

When $x = 0$, $y = 3$, so $3 = m(0) + c \implies c = 3$.

Use another point, e.g., $(1, 5)$ , and $c=3$ to find $m$ .
$5 = m(1) + 3 \implies 5 - 3 = m \implies m = 2$.
Write the rule.
The rule is $\boxed{y = 2x + 3}$.
12. Given $x = \{1, 2, 3, 4\}$ and $y = \{5, 7, 9, 11\}$ :

Determine the slope ( $m$ ) by finding the change in $y$ over the change in $x$ .

The difference in $y$ values is constant: $7-5=2$, $9-7=2$, $11-9=2$.  
The difference in $x$ values is constant: $2-1=1$, $3-2=1$, $4-3=1$.  
So, $m = \frac{\Delta y}{\Delta x} = \frac{2}{1} = 2$.

Use one of the pairs and the slope to find the y-intercept ( $c$ ) in $y = mx + c$ .
Using $(1, 5)$: $5 = 2(1) + c \implies 5 = 2 + c \implies c = 3$.
Write the rule and verify.
The rule is $\boxed{y = 2x + 3}$. Verification: For $x = 2$: $y = 2(2) + 3 = 4 + 3 = 7$. (Matches the given $y$ value) For $x = 4$: $y = 2(4) + 3 = 8 + 3 = 11$. (Matches the given $y$ value)
13. A linear mapping is characterized by a constant difference in the output values for a constant difference in the input values. Its graph is a straight line. An example is $y = 3x + 1$ , where for every unit increase in $x$ , $y$ increases by 3. An exponential mapping is characterized by a constant ratio (or multiplicative factor) in the output values for a constant difference in the input values. Its graph is a curve that either grows or decays rapidly. An example is $y = 2^x$ , where for every unit increase in $x$ , $y$ is multiplied by 2.
1. Given the set of ordered pairs: $(1, 2), (2, 4), (3, 8), (4, 16)$ .

Check for a constant difference in $y$ values (linear).

$4 - 2 = 2$  
$8 - 4 = 4$  
$16 - 8 = 8$  
The differences are not constant, so it is not a linear mapping.

Check for a constant ratio in $y$ values (exponential).

$\frac{4}{2} = 2$  
$\frac{8}{4} = 2$  
$\frac{16}{8} = 2$  
The ratios are constant.

Justify the answer.

This mapping is **exponential**. This is because for a constant increase in the input ($x$ increases by 1 each time), the output ($y$) is multiplied by a constant factor of 2.

15. Given the linear mapping $y = 3x + 2$ :

Calculate output values for $x = 0, 1, 2, 3$ .

For $x = 0: y = 3(0) + 2 = 2$  
For $x = 1: y = 3(1) + 2 = 5$  
For $x = 2: y = 3(2) + 2 = 8$  
For $x = 3: y = 3(3) + 2 = 11$

Organize results in a table.
$$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 2 \\ 1 & 5 \\ 2 & 8 \\ 3 & 11 \\ \hline \end{array} $$
16. The mobile phone plan charges a fixed fee of GH₵10 plus GH₵0.50 per minute. Let $x$ represent minutes and $y$ represent total cost. The rate per minute is the slope ( $m$ ), so $m = 0.50$ . The fixed fee is the y-intercept ( $c$ ), so $c = 10$ . The rule for this mapping is $\boxed{y = 0.50x + 10}$ .
1. Given: Domain: $2, 4, 6, 8$ and Co-domain: $7, 13, 19, 25$ .

Identify constant differences.

Constant difference in domain ($\Delta x$): $4-2=2$, $6-4=2$, $8-6=2$. So, $\Delta x = 2$.  
Constant difference in co-domain ($\Delta y$): $13-7=6$, $19-13=6$, $25-19=6$. So, $\Delta y = 6$.

Determine the rule $y = mx + c$ .

The slope $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$.  
Using the first pair $(2, 7)$ and $m=3$:  
$7 = 3(2) + c$  
$7 = 6 + c$  
$c = 1$  
The rule is $\boxed{y = 3x + 1}$.

18. For the mapping $y = 2x + 1$ :

Create a table of values for $x = -2, -1, 0, 1, 2$ .

For $x = -2: y = 2(-2) + 1 = -4 + 1 = -3$  
For $x = -1: y = 2(-1) + 1 = -2 + 1 = -1$  
For $x = 0: y = 2(0) + 1 = 0 + 1 = 1$  
For $x = 1: y = 2(1) + 1 = 2 + 1 = 3$  
For $x = 2: y = 2(2) + 1 = 4 + 1 = 5$
$$
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -3 \\
-1 & -1 \\
0 & 1 \\
1 & 3 \\
2 & 5 \\
\hline
\end{array}
$$

Explain what $m$ and $c$ represent.
In the rule $y = 2x + 1$:
- $m = 2$ represents the slope or the rate of change. It indicates that for every unit increase in $x$ , the value of $y$ increases by 2.
- $c = 1$ represents the y-intercept. It is the value of $y$ when $x$ is 0.
1. Given the rule $y = -2x + 5$ :
Determine the output when $x = 3$ .
$y = -2(3) + 5$ $y = -6 + 5$ $y = -1$

Explain what this ordered pair represents.

The ordered pair is $(3, -1)$. This represents a point on the graph of the mapping where an input of 3 results in an output of $-1$.

20. A taxi charges GH₵2.50 per kilometre plus a fixed charge of GH₵5.

Write the mapping rule.

Let $x$ be the number of kilometres and $y$ be the total cost.  
The charge per kilometre is the slope ($m$), so $m = 2.50$.  
The fixed charge is the y-intercept ($c$), so $c = 5$.  
The mapping rule is $\boxed{y = 2.50x + 5}$.

Calculate the cost for a 15 km journey.

Substitute $x = 15$ into the rule:  
$y = 2.50(15) + 5$  
$y = 37.50 + 5$  
$y = 42.50$  
The cost for a 15 km journey is $\boxed{\text{GH₵}42.50}$.

Step-by-step solution

Use the point $(0, 3)$ to find $c$ .

Use another point, e.g., $(1, 5)$ , and $c=3$ to find $m$ .

Write the rule.

Determine the slope ( $m$ ) by finding the change in $y$ over the change in $x$ .

Use one of the pairs and the slope to find the y-intercept ( $c$ ) in $y = mx + c$ .

Write the rule and verify.

Check for a constant difference in $y$ values (linear).

Check for a constant ratio in $y$ values (exponential).

Justify the answer.

Calculate output values for $x = 0, 1, 2, 3$ .

Organize results in a table.

Identify constant differences.

Determine the rule $y = mx + c$ .

Create a table of values for $x = -2, -1, 0, 1, 2$ .

Explain what $m$ and $c$ represent.

Determine the output when $x = 3$ .

Explain what this ordered pair represents.

Write the mapping rule.

Calculate the cost for a 15 km journey.

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More Computer Science Questions

Step-by-step solution

Use the point (0,3)(0, 3)(0,3) to find ccc.

Use another point, e.g., (1,5)(1, 5)(1,5), and c=3c=3c=3 to find mmm.

Write the rule.

Determine the slope (mmm) by finding the change in yyy over the change in xxx.

Use one of the pairs and the slope to find the y-intercept (ccc) in y=mx+cy = mx + cy=mx+c.

Write the rule and verify.

Check for a constant difference in yyy values (linear).

Check for a constant ratio in yyy values (exponential).

Justify the answer.

Calculate output values for x=0,1,2,3x = 0, 1, 2, 3x=0,1,2,3.

Organize results in a table.

Identify constant differences.

Determine the rule y=mx+cy = mx + cy=mx+c.

Create a table of values for x=−2,−1,0,1,2x = -2, -1, 0, 1, 2x=−2,−1,0,1,2.

Explain what mmm and ccc represent.

Determine the output when x=3x = 3x=3.

Explain what this ordered pair represents.

Write the mapping rule.

Calculate the cost for a 15 km journey.

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More Computer Science Questions

Use the point $(0, 3)$ to find $c$ .

Use another point, e.g., $(1, 5)$ , and $c=3$ to find $m$ .

Determine the slope ( $m$ ) by finding the change in $y$ over the change in $x$ .

Use one of the pairs and the slope to find the y-intercept ( $c$ ) in $y = mx + c$ .

Check for a constant difference in $y$ values (linear).

Check for a constant ratio in $y$ values (exponential).

Calculate output values for $x = 0, 1, 2, 3$ .

Determine the rule $y = mx + c$ .

Create a table of values for $x = -2, -1, 0, 1, 2$ .

Explain what $m$ and $c$ represent.

Determine the output when $x = 3$ .