Question

which of these relations is a function? (followed by four graphs: first with two lines to a point, second a curve, third a circle, fourth a circle tangent to x - axis)

Explanation:

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

Analyzing each graph:

1. First graph (two lines to a point): A vertical line would intersect at most once (since it’s two straight segments meeting at a point). Passes the vertical line test.
2. Second graph (curved line): The curve does not fail the vertical line test (no vertical line intersects it more than once). Passes.
3. Third graph (circle): A vertical line through the circle’s diameter will intersect twice. Fails the vertical line test.
4. Fourth graph (circle): A vertical line through the circle will intersect twice. Fails the vertical line test.

However, re - evaluating the first graph: If it’s two distinct lines (not a single relation with one input - output), but typically, the first graph (with two straight segments from left to right, meeting at a point) and the second graph (the curve) are candidates. Wait, the first graph: if we consider the definition of a function (each \( x \) has at most one \( y \)), in the first graph, for any vertical line, there is at most one intersection. The second graph also has each \( x \) with one \( y \). But the third and fourth (circles) fail.

But looking at the standard "which is a function" questions, the first graph (the two straight segments) and the second graph (the curve) are functions. But maybe the first graph is two rays or segments, and the second is a curve. Wait, the key is the vertical line test.

Wait, the first graph: let's assume the first graph is two lines (maybe a piece - wise linear function) and the second is a curve. The circles (third and fourth) are not functions.

So the answer would be the first and second graphs. But since the user provided the images, and in typical problems, the first graph (with two straight lines) and the second graph (the curve) are functions. If we have to choose one, maybe the first or the second. But likely, the first graph (the two straight segments) is a function, and the second (the curve) is also a function. The circles are not.

Answer:

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

Analyzing each graph:

1. First graph (two lines to a point): A vertical line would intersect at most once (since it’s two straight segments meeting at a point). Passes the vertical line test.
2. Second graph (curved line): The curve does not fail the vertical line test (no vertical line intersects it more than once). Passes.
3. Third graph (circle): A vertical line through the circle’s diameter will intersect twice. Fails the vertical line test.
4. Fourth graph (circle): A vertical line through the circle will intersect twice. Fails the vertical line test.

However, re - evaluating the first graph: If it’s two distinct lines (not a single relation with one input - output), but typically, the first graph (with two straight segments from left to right, meeting at a point) and the second graph (the curve) are candidates. Wait, the first graph: if we consider the definition of a function (each \( x \) has at most one \( y \)), in the first graph, for any vertical line, there is at most one intersection. The second graph also has each \( x \) with one \( y \). But the third and fourth (circles) fail.

But looking at the standard "which is a function" questions, the first graph (the two straight segments) and the second graph (the curve) are functions. But maybe the first graph is two rays or segments, and the second is a curve. Wait, the key is the vertical line test.

Wait, the first graph: let's assume the first graph is two lines (maybe a piece - wise linear function) and the second is a curve. The circles (third and fourth) are not functions.

So the answer would be the first and second graphs. But since the user provided the images, and in typical problems, the first graph (with two straight lines) and the second graph (the curve) are functions. If we have to choose one, maybe the first or the second. But likely, the first graph (the two straight segments) is a function, and the second (the curve) is also a function. The circles are not.