Which graph represents a pair of inverse functions?

Question

Which graph represents a pair of inverse functions?
(1 point)
Responses

Two graphs are shown on a coordinate grid.The first graph is a curve opening up that passes through left-parenthesis negative 2 comma 4 right-parenthesis, left-parenthesis negative 1 comma 1 right-parenthesis, left-parenthesis 0 comma 0 right-parenthesis, left-parenthesis 1 comma 1 right-parenthesis, and left-parenthesis 2 comma 4 right-parenthesis.

The second graph is a curve opening to the right that passes through left-parenthesis 4 comma negative 2 right-parenthesis, left-parenthesis 1 comma negative 1 right-parenthesis, left-parenthesis 0 comma 0 right-parenthesis, left-parenthesis 1 comma 1 right-parenthesis, and left-parenthesis 4 comma 2 right-parenthesis.
Image with alt text: Two graphs are shown on a coordinate grid. The first graph is a curve opening up that passes through left-parenthesis negative 2 comma 4 right-parenthesis, left-parenthesis negative 1 comma 1 right-parenthesis, left-parenthesis 0 comma 0 right-parenthesis, left-parenthesis 1 comma 1 right-parenthesis, and left-parenthesis 2 comma 4 right-parenthesis. The second graph is a curve opening to the right that passes through left-parenthesis 4 comma negative 2 right-parenthesis, left-parenthesis 1 comma negative 1 right-parenthesis, left-parenthesis 0 comma 0 right-parenthesis, left-parenthesis 1 comma 1 right-parenthesis, and left-parenthesis 4 comma 2 right-parenthesis.

Two graphs are shown on a coordinate grid.The first graph starts at the left of the graph and decreases passing through the points left-parenthesis negative 9 comma 3 right-parenthesis, left-parenthesis negative 4 comma 2 right-parenthesis and left-parenthesis negative 1 comma 1 and ends at left parenthesis 0 comma 0 right-parenthesis.

The second graph begins at left-parenthesis 0 comma 0 right-parenthesis and increases forever passing through left-parenthesis 1 comma 1 right-parenthesis, left-parentheis 4 comma 2 right-parenthesis, and left-parenthesis 9 comma 3 right-parenthesis.
Image with alt text: Two graphs are shown on a coordinate grid. The first graph starts at the left of the graph and decreases passing through the points left-parenthesis negative 9 comma 3 right-parenthesis, left-parenthesis negative 4 comma 2 right-parenthesis and left-parenthesis negative 1 comma 1 and ends at left parenthesis 0 comma 0 right-parenthesis. The second graph begins at left-parenthesis 0 comma 0 right-parenthesis and increases forever passing through left-parenthesis 1 comma 1 right-parenthesis, left-parentheis 4 comma 2 right-parenthesis, and left-parenthesis 9 comma 3 right-parenthesis.

Two graphs are shown on a coordinate grid.The first graph starts just above the x-axis and increases forever through left-parenthesis negative 1 comma one half right-parenthesis, left-parenthesis 0 comma 1 right-parenthesis, left-parenthesis 1 comma 2 right-parenthesis, and left-parenthesis 2 comma 4 right-parenthesis.

The second graph starts just to the right of the y-axis and increases forever through left-parenthesis one half comma negative 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma 1 right-parenthesis, left-parenthesis 4 comma 2 right-parenthesis, and left-parenthesis 8 comma 3 right parenthesis.
Image with alt text: Two graphs are shown on a coordinate grid. The first graph starts just above the x-axis and increases forever through left-parenthesis negative 1 comma one half right-parenthesis, left-parenthesis 0 comma 1 right-parenthesis, left-parenthesis 1 comma 2 right-parenthesis, and left-parenthesis 2 comma 4 right-parenthesis. The second graph starts just to the right of the y-axis and increases forever through left-parenthesis one half comma negative 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma 1 right-parenthesis, left-parenthesis 4 comma 2 right-parenthesis, and left-parenthesis 8 comma 3 right parenthesis.

Two graphs are shown on a coordinate grid.The first graph starts just to the right of the y-axis and increases forever through left-parenthesis one half comma negative 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma 1 right-parenthesis, left-parenthesis 4 comma 2 right-parenthesis, and left-parenthesis 8 comma 3 right parenthesis.

The second graph begins just to the right of the y-axis and decreases forever passing through left-parenthesis one half comma 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis 4 comma negative 2 right-parenthesis, and left-parenthesis 8 comma negative 3 right-parenthesis.
Image with alt text: Two graphs are shown on a coordinate grid. The first graph starts just to the right of the y-axis and increases forever through left-parenthesis one half comma negative 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma 1 right-parenthesis, left-parenthesis 4 comma 2 right-parenthesis, and left-parenthesis 8 comma 3 right parenthesis. The second graph begins just to the right of the y-axis and decreases forever passing through left-parenthesis one half comma 1 right-parenthesis, left-parenthesis 1 comma 0 right-parenthesis, left-parenthesis 2 comma negative 1 right-parenthesis, left-parenthesis 4 comma negative 2 right-parenthesis, and left-parenthesis 8 comma negative 3 right-parenthesis.
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GPT-4o mini · Answer

To determine which pair of graphs represents inverse functions, we should check if each point on one graph corresponds to a point on the other graph when the coordinates are swapped (i.e., if point \( (a, b) \) on the first graph corresponds to point \( (b, a) \) on the second graph).

Let’s analyze the given graph pairs:

Option 1:

Graph 1: Passes through points \( (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) \).
Graph 2: Passes through points \( (4, -2), (1, -1), (0, 0), (1, 1), (2, 4) \).
This option does not exhibit the property of inverse functions, as the points do not correspond by swapping coordinates.

Option 2:

Graph 1: Passes through points \( (-9, 3), (-4, 2), (-1, 1), (0, 0) \).
Graph 2: Passes through points \( (0, 0), (1, 1), (4, 2), (9, 3) \).
This option shows that if you take a point from Graph 1 (for example, \( (0, 0) \)), it corresponds to a point in Graph 2, but others do not reflect the nature of inverse functions well.

Option 3:

Graph 1: Increases through points \( (-1, 0.5), (0, 1), (1, 2), (2, 4) \).
Graph 2: Increases through points \( (0.5, -1), (1, 0), (2, 1), (4, 2), (8, 3) \).
This does not appear to represent inverse functions.

Option 4:

Graph 1: Increases through points \( (0.5, -1), (1, 0), (2, 1), (4, 2), (8, 3) \).
Graph 2: Decreases through points \( (0.5, 1), (1, 0), (2, -1), (4, -2), (8, -3) \).
The key here is to notice that both graphs pass through (1,0) and the other points do not represent a proper inverse.

Conclusion

The best match for a pair of inverse functions is Option 4, where the coordinates in the pair reflect each other (i.e., points \( (a, b) \) and \( (b, a) \)). Therefore, the response is option 4.