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 (ellipse)

Draw a vertical line through the ellipse. It will intersect the ellipse at two points. So, it fails the vertical line test and is not a function.

Step3: Analyze the second graph (horizontal line)

A horizontal line has the equation \( y = k \) (where \( k \) is a constant). Any vertical line \( x = a \) will intersect this horizontal line at most once (at \( (a, k) \)). So, it passes the vertical line test and is a function.

Step4: Analyze the third graph (V - shaped with a vertex on the x - axis, opening to the left and down)

Draw a vertical line. It will intersect this graph at two points. So, it fails the vertical line test and is not a function.

Step5: Analyze the fourth graph (a piece - wise linear graph)

For any vertical line \( x = a \), it will intersect this graph at most once. So, it passes the vertical line test and is a function.

Answer:

The second graph (horizontal line) and the fourth graph (the piece - wise linear graph) are functions.