Question

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

Explanation:

To determine if a relation is a function, we use the vertical line test: a graph represents a function if no vertical line intersects the graph more than once.

Step 1: Analyze the first graph (top - left)

Check each \( x \) - value. Each \( x \) has at most one \( y \) - value (the points and segments don't have multiple \( y \) - values for the same \( x \)). So it passes the vertical line test.

Step 2: Analyze the second graph (top - right)

This graph is a curve that loops. A vertical line can intersect this curve at two points (for example, a vertical line through the middle of the loop). So it fails the vertical line test.

Step 3: Analyze the third graph (bottom - left)

This is a circle. A vertical line through the circle will intersect it at two points (since a circle has the equation \( (x - h)^2+(y - k)^2 = r^2 \), for a given \( x \), there are two \( y \) - values). So it fails the vertical line test.

Step 4: Analyze the fourth graph (bottom - right)

For any vertical line, it will intersect this graph at most once (the graph is a smooth curve with no vertical line intersections at more than one point). So it passes the vertical line test.

Answer:

The first (top - left) and the fourth (bottom - right) graphs represent functions.