Inverse Function Worksheets - Free Printable

Problem Analysis:

The task involves analyzing graphs to determine whether they represent functions and checking for the existence of their inverses. Additionally, we need to identify which graph does not have an inverse.

#### Part A: Determine if each graph represents a function and check for the existence of its inverse.
To solve this, we will use two key concepts:
1. Vertical Line Test: A graph represents a function if every vertical line intersects the graph at most once.
2. Horizontal Line Test: A function has an inverse if every horizontal line intersects the graph at most once. This ensures that the function is one-to-one (injective).

#### Part B: Identify which graph does not have an inverse.
Using the Horizontal Line Test, we will determine which graph fails to be one-to-one.

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Solution:

#### Graph 1:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Every horizontal line intersects the graph at most once. Therefore, it has an inverse.
- Conclusion: Function, Inverse Exists.

#### Graph 2:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Some horizontal lines intersect the graph more than once. Therefore, it does not have an inverse.
- Conclusion: Function, No Inverse.

#### Graph 3:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Every horizontal line intersects the graph at most once. Therefore, it has an inverse.
- Conclusion: Function, Inverse Exists.

#### Graph 4:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Every horizontal line intersects the graph at most once. Therefore, it has an inverse.
- Conclusion: Function, Inverse Exists.

#### Graph 5:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Every horizontal line intersects the graph at most once. Therefore, it has an inverse.
- Conclusion: Function, Inverse Exists.

#### Graph 6:
- Vertical Line Test: Every vertical line intersects the graph at most once. Therefore, it is a function.
- Horizontal Line Test: Some horizontal lines intersect the graph more than once. Therefore, it does not have an inverse.
- Conclusion: Function, No Inverse.

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Part B: Which graph does not have an inverse?

From the analysis above, Graph 2 and Graph 6 fail the Horizontal Line Test. Therefore, they do not have inverses.

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Final Answer:

\[
\boxed{\text{Graph 2 and Graph 6}}
\]