Question

check your understanding

1. state which relations are functions. explain.

a) {(-5, 1), (-3, 2), (-1, 3), (1, 2)}
b) mapping diagram: left set: {-1, 1, 3, 5}; right set: {-3, -1, 0, 2}; arrows: -1 → -3, -1 → -1, 1 → 0, 3 → 2, 5 → 2
c) {(0, 4), (3, 5), (5, -2), (0, 1)}
d) mapping diagram: left set: {-4, -2, 2, 5}; right set: {1, 3, 6}; arrows: -4 → 1, -4 → 3, -2 → 1, 2 → 1, 5 → 6

1. use a ruler and the vertical-line test to determine which graphs are functions.

a) graph with points: (-4, 2), (-3, 2), (-2, 2), (-1, 2); (1, -3), (2, -3), (3, -2), (4, -1)
b) graph: a curve (horizontal parabola) passing through (-2, 2), (0, 0), (4, 4)
c) graph: a parabola opening upward, vertex at (0, -2), passing through (-2, 0), (2, 0)
d) graph: a circle centered at (-2, 0) with radius 2
e) graph: a v - shaped graph (absolute value) with vertex at (0, 2), left arm from (-4, -2) to (0, 2), right arm from (0, 2) to (4, -2)
f) graph: two lines: one from (-4, 4) to (2, 0), another from (-4, -4) to (2, 0)

Explanation:

Problem 1: Determine which relations are functions

Part (a)

Step 1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value).
For the set \(\{(-5,1),(-3,2),(-1,3),(1,2)\}\), we check the x - values: \(-5\), \(-3\), \(-1\), \(1\). Each x - value appears only once, so each input has exactly one output.

Step 1: Analyze the mapping

The input \(1\) is mapped to \(-3\), \(-1\), and \(0\). In a function, each input must have exactly one output. Since the input \(1\) has multiple outputs, this relation does not satisfy the definition of a function.

Step 1: Check the x - values

For the set \(\{(0,4),(3,5),(5, - 2),(0,1)\}\), the x - value \(0\) is paired with \(4\) and \(1\). A function requires that each x - value has exactly one y - value. Since \(0\) has two different y - values, this relation is not a function.

Answer:

This relation is a function because each x - value has exactly one y - value.

Part (b)