Question

which of the following sets of data represent valid functions?
$f = \\{(-1, -2), (1, 3), (1, 5), (9, 7), (11, 14)\\}$
$g = \\{(-4, -5), (3, 2), (4, 6), (8, 7), (15, 16)\\}$
$r = \\{(0, -2), (3, 1), (6, 4), (0, -2), (16, 14)\\}$
$s = \\{(-1, -4), (1, 3), (5, 4), (9, 7), (14, 11)\\}$
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Explanation:

Step1: Recall function definition

A function is a relation where each input (x-value) has exactly one output (y-value).

Step2: Check set F

Input $x=1$ maps to $y=3$ and $y=5$.
$F$ has repeated x with different y: invalid.

Step3: Check set G

All x-values $\{-4,3,4,8,15\}$ are unique, each maps to one y.
$G$ has no repeated x: valid.

Step4: Check set R

Input $x=0$ maps to $y=-2$ twice, but it's the same y-value.
$R$ has repeated x with same y: valid.

Step5: Check set S

All x-values $\{-1,1,5,9,14\}$ are unique, each maps to one y.
$S$ has no repeated x: valid.

Answer:

G = {(-4, -5), (3, 2), (4, 6), (8, 7), (15, 16)}, R = {(0, -2), (3, 1), (6, 4), (0, -2), (16, 14)}, S = {(-1, -4), (1, 3), (5, 4), (9, 7), (14, 11)}