Ordered Pairs
Table
Mapping
Graph
Domain & Range
100
Given the relation R = {(1,2), (3,4), (5,6)}, determine if this is a function.
Answer: Yes, this is a function.
Reasoning: In this relation, each first element (domain) appears exactly once and maps to exactly one second element (range). There are no repeated x-values, so it satisfies the definition of a function where each input has exactly one output
100
Looking at the table below, is this relation a function?
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
Yes, this is a function because:
- Each x-value appears exactly once
- Each x-value is paired with exactly one y-value
- The pattern shows y = 2x
100
In the mapping below, is this a function? Why or why not?
Set A = {1, 2, 3}
Set B = {a, b}
Mappings: 1 → a, 2 → b, 3 → a
Yes, this is a function because:
- Each element in Set A (domain) is mapped to exactly one element in Set B (codomain)
- It's okay that both 1 and 3 map to 'a' (multiple elements can map to the same output)
- There are no elements in Set A left unmapped
100
Looking at the graph below, determine if this relation is a function. Explain why or why not.
No, this is not a function. Looking at the graph, there are two y-values (points) for x = -1. This violates the vertical line test - a vertical line intersects the relation at more than one point, meaning one input has multiple outputs.
100
Consider the relation R = {(1,2), (3,4), (5,6)}
What is the domain and range of this relation?
Domain = {1, 3, 5}
Range = {2, 4, 6}
Explanation: The domain consists of all first elements (x-values), while the range consists of all second elements (y-values) in the ordered pairs.
200
Given the relation R = {(1,2), (1,3), (2,4)}, determine if this is a function.
No, this is not a function.
Reasoning: The element 1 in the domain maps to both 2 and 3. In a function, each input can only map to exactly one output. This violates the definition of a function.
200
Is this relation a function?
| x | y |
|---|---|
| 1 | 3 |
| 1 | 4 |
| 2 | 5 |
No, this is not a function because:
- The x-value 1 is paired with two different y-values (3 and 4)
- A function must have exactly one output for each input
200
onsider the relation R from set P = {1, 2} to set Q = {x, y, z}
where 1 → x and 1 → y. Is this a function? Explain.
No, this is not a function because:
- Element 1 maps to two different values (x and y)
- Functions must have exactly one output for each input
- This is a relation but fails the function test due to multiple outputs
200
In the graph, we see a semi-circle. Is this relation a function? Explain using the vertical line test.
Yes, this is a function. When applying the vertical line test, any vertical line will intersect the semi-circle at most once. This means each x-value corresponds to exactly one y-value, satisfying the definition of a function.
200
Consider the relation R = {(2,4), (2,6), (3,9), (4,16)}
Is this relation a function? What is its domain and range?
No, this is not a function because 2 maps to both 4 and 6.
Domain = {2, 3, 4}
Range = {4, 6, 9, 16}
Explanation: A function must have exactly one output for each input. Here, 2 has two outputs.
300
Given sets A = {1, 2, 3} and B = {x, y}, is it possible to create a function f from A to B? If not, explain why.
Yes, it is possible to create a function, but it would not be a one-to-one function.
Reasoning: Since A has 3 elements and B has 2 elements, some elements in B must be used more than once as outputs. For example, f = {(1,x), (2,x), (3,y)} would be a valid function from A to B.
300
In the table below, what makes this a relation but not a function?
| x | y |
|---|---|
| 2 | 7 |
| 4 | 7 |
| 2 | 9 |
This is only a relation because:
- It violates the function rule in two ways:
1. The x-value 2 appears twice (with y=7 and y=9)
2. Two different x-values (2 and 4) share the same y-value (7)
- A relation can have these properties, but a function cannot
300
Given sets X = {a, b, c} and Y = {1, 2}
Which of these mappings represents a function?
i) a → 1, b → 2, c → 1
ii) a → 1, b → 1
iii) a → 1, b → 2, c → 1, c → 2
Only mapping (i) represents a function because:
- It maps every element of set X to exactly one element in set Y
- Mapping (ii) is not a function as 'c' has no mapping
- Mapping (iii) is not a function as 'c' maps to two values
300
Looking at the graph, which shows a circle centered at the origin, explain why this is not a function and identify the x-values where it fails the vertical line test.
This is not a function. The vertical line test fails at every x-value between -1 and 1 (except at the endpoints). For example, at x = 0, the vertical line intersects the circle at two points: (0,1) and (0,-1). This means one input has two outputs, violating the definition of a function.
300
The relation R is defined as all ordered pairs (x,y) where y = √x for real numbers.
What is the domain and range of this relation? Express using interval notation.
Domain = [0,∞)
Range = [0,∞)
Explanation: Since we can't take the square root of negative numbers in the real number system, x must be greater than or equal to 0. The square root of a non-negative number is always non-negative.
400
Which of these relations is NOT a function?
R1 = {(1,2), (1,3), (2,4)}
R2 = {(0,0), (1,1), (2,2)}
R3 = {(a,b), (b,c), (c,d)}
R1 is not a function
Explanation: In R1, the input value 1 maps to both 2 and 3, violating the definition of a function where each input must map to exactly one output.
400
The table shows test scores for different students. Is this a function if x represents students and y represents scores?
| Student (x) | Score (y) |
|-------------|-----------|
| Alice | 85 |
| Bob | 92 |
| Charlie | 85 |
| David | 78 |
Yes, this is a function because:
- Each student (input) appears exactly once
- Each student has exactly one score (output)
- It doesn't matter that two students have the same score (85)
- Functions can have different inputs mapping to the same output
400
Is every function a relation? Is every relation a function? Explain using a simple mapping example.
Every function is a relation, but not every relation is a function
Example mapping to demonstrate:
Set A = {1, 2}
Set B = {p, q}
Relation 1: 1 → p, 2 → q (This is both a relation and a function)
Relation 2: 1 → p, 1 → q (This is a relation but not a function)
The second relation fails to be a function because one input maps to multiple outputs.
400
The graph shows a parabola. Explain why this is a function and describe what makes parabolas special in terms of the vertical line test.
This is a function because any vertical line will intersect the parabola exactly once. Parabolas are special because they represent quadratic functions where each x-value corresponds to exactly one y-value. The "U" shape ensures that no matter where you draw your vertical line, it will only cross the curve once.
400
Consider the relation R = {(x,y) | x² + y² = 25}
Is this a function? What is its domain and range?
Not a function (circle equation - each x can have two y values)
Domain = [-5,5]
Range = [-5,5]
Explanation: This is the equation of a circle with radius 5. For most x-values in the domain (except ±5), there are two corresponding y-values.
500
Given two relations:
R₁ = {(1,2), (2,3), (3,4)}
R₂ = {(1,2), (2,3), (3,4), (4,5)}
Are both relations functions? If yes, what's the main difference between them?
Yes, both are functions.
Reasoning: In both relations, each domain element maps to exactly one range element. The main difference is their domains
500
Given the table below, which statement is correct?
| x | y |
|---|---|
| 5 | 1 |
| 3 | 1 |
| 4 | 1 |
This is a valid relation
This is a valid function
Both statements are true:
- This is a valid relation
- This is a valid function because:
* Each x-value appears exactly once
* Each x-value pairs with exactly one y-value
500
Given the mapping diagram below, determine if it represents a function or just a relation. If it's not a function, explain what changes would make it a function.
Set M = {4, 5, 6}
Set N = {7, 8, 9}
Mappings: 4 → 7, 5 → 8, 5 → 9, 6 → 7
This is not a function because:
- Element 5 maps to both 8 and 9, violating the function rule of one output per input
To make it a function, either:
- Remove the mapping 5 → 9, leaving 5 → 8
OR
- Remove the mapping 5 → 8, leaving 5 → 9
Either change would result in a valid function where each input has exactly one output.
500
The graph shows two intersecting lines. Is this a function? If not, identify all points where it fails the vertical line test and explain why these points are significant.
No, this is not a function. The two intersecting lines fail the vertical line test at multiple x-values. Most notably, at the x-coordinate of the intersection point, there are two different y-values (except at the point where the lines cross). This relation represents two linear functions combined, but together they don't form a single function because some x-values map to two different y-values.
500
Given the relation R = {(x,y) | y = (x-2)/(x+3), where x is a real number}
What is the domain and range of this relation? Express using interval notation.
Domain = (-∞,-3) ∪ (-3,∞)
Range = (-∞,1) ∪ (1,∞)
Explanation: Domain excludes x = -3 because denominator cannot be zero. Range excludes y = 1 because as x approaches ±∞, y approaches 1 but never equals it.