Question

5 a. complete the graph by plotting (input, output) pairs for input values of 3, 4, and 5.
input: x, a whole number
output: y, all prime numbers less than or equal to x
b. is y a function of x? how can you tell by looking at the graph?

Explanation:

Step1: Find y for x=3

Prime numbers ≤3: 2,3 → Count: $y=2$
Pair: $(3, 2)$

Step2: Find y for x=4

Prime numbers ≤4: 2,3 → Count: $y=2$
Pair: $(4, 2)$

Step3: Find y for x=5

Prime numbers ≤5: 2,3,5 → Count: $y=3$
Pair: $(5, 3)$

Step4: Analyze function definition

A function has one output per input.

Step5: Apply vertical line test

No vertical line intersects >1 graph point.

Answer:

a. Plot the points $(3, 2)$, $(4, 2)$, and $(5, 3)$ on the given coordinate grid (along with the existing $(2, 2)$ point).
b. Yes, $y$ is a function of $x$. Using the vertical line test, any vertical line drawn on the graph will intersect the set of plotted points at most once, which satisfies the definition of a function (each input has exactly one output).