Question

1. determine whether each graph represents the graph of a function. if so, determine whether the graph is the graph of a one - to - one function.

(a)
(b)
(c)
(d)

Explanation:

To determine if a graph represents a function, we use the vertical line test: a graph is a function if no vertical line intersects it more than once. To determine if a function is one - to - one, we use the horizontal line test: a function is one - to - one if no horizontal line intersects its graph more than once.

Part (a)

Step 1: Vertical Line Test

If we draw a vertical line (parallel to the \(y\) - axis) anywhere on the graph, it will intersect the graph at more than one point. So, by the vertical line test, the graph in (a) does not represent a function.

Step 2: Conclusion for One - to - One

Since it is not a function, we do not need to check for one - to - one.

Part (b)

Step 1: Vertical Line Test

For the parabola - shaped graph (opening downwards), any vertical line will intersect the graph at most once. So, by the vertical line test, this graph represents a function.

Step 2: Horizontal Line Test

If we draw a horizontal line (parallel to the \(x\) - axis), it will intersect the graph of the downward - opening parabola at two points (except at the vertex). So, by the horizontal line test, the function is not one - to - one.

Part (c)

Step 1: Vertical Line Test

If we draw a vertical line, it will intersect the graph at more than one point. So, by the vertical line test, the graph in (c) does not represent a function.

Step 2: Conclusion for One - to - One

Since it is not a function, we do not need to check for one - to - one.

Part (d)

Step 1: Vertical Line Test

Any vertical line drawn on this graph will intersect it at most once. So, by the vertical line test, the graph represents a function.

Step 2: Horizontal Line Test

Any horizontal line drawn on this graph will intersect it at most once. So, by the horizontal line test, the function is one - to - one.

Final Answers:

• (a): Not a function.
• (b): Function, but not one - to - one.
• (c): Not a function.
• (d): Function and one - to - one.

Answer:

To determine if a graph represents a function, we use the vertical line test: a graph is a function if no vertical line intersects it more than once. To determine if a function is one - to - one, we use the horizontal line test: a function is one - to - one if no horizontal line intersects its graph more than once.

Part (a)

Step 1: Vertical Line Test

If we draw a vertical line (parallel to the \(y\) - axis) anywhere on the graph, it will intersect the graph at more than one point. So, by the vertical line test, the graph in (a) does not represent a function.

Step 2: Conclusion for One - to - One

Since it is not a function, we do not need to check for one - to - one.

Part (b)

Step 1: Vertical Line Test

For the parabola - shaped graph (opening downwards), any vertical line will intersect the graph at most once. So, by the vertical line test, this graph represents a function.

Step 2: Horizontal Line Test

If we draw a horizontal line (parallel to the \(x\) - axis), it will intersect the graph of the downward - opening parabola at two points (except at the vertex). So, by the horizontal line test, the function is not one - to - one.

Part (c)

Step 1: Vertical Line Test

If we draw a vertical line, it will intersect the graph at more than one point. So, by the vertical line test, the graph in (c) does not represent a function.

Step 2: Conclusion for One - to - One

Since it is not a function, we do not need to check for one - to - one.

Part (d)

Step 1: Vertical Line Test

Any vertical line drawn on this graph will intersect it at most once. So, by the vertical line test, the graph represents a function.

Step 2: Horizontal Line Test

Any horizontal line drawn on this graph will intersect it at most once. So, by the horizontal line test, the function is one - to - one.

Final Answers:

• (a): Not a function.
• (b): Function, but not one - to - one.
• (c): Not a function.
• (d): Function and one - to - one.