Question

which of these relations are functions? select all that apply.

Explanation:

Step1: Recall the vertical line test

A relation is a function if no vertical line intersects its graph more than once.

Step2: Analyze the first graph (circle)

A vertical line through the circle will intersect it at two points (e.g., the y - axis intersects the top and bottom of the circle). So, it is not a function.

Step3: Analyze the second graph (piece - wise linear)

For any vertical line, it will intersect this graph at most once. So, this relation is a function.

Step4: Analyze the third graph (S - shaped curve)

A vertical line will intersect this curve at more than one point (since it crosses over itself vertically). So, it is not a function.

Step5: Analyze the fourth graph (increasing curve)

For any vertical line, it will intersect this graph at most once. So, this relation is a function.

Answer:

The second graph (the piece - wise linear graph) and the fourth graph (the increasing curve graph) are functions.