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To determine which functions have an inverse from the given tables, we need to check if each function is one-to-one. A function is one-to-one if it assigns unique outputs to unique inputs. In other words, the function must not repeat any output values for different input values.

Let's analyze each table:

### Table (a) \( g(x) \)
| \( g(x) \) | \( x \)  |
|------------|----------|
| -2         | -9       |
| -1         | -3       |
| 0          | 1        |
| 1          | 3        |
| 2          | 9        |
| 3          | 19       |

- Here, every \( g(x) \) value is unique. Therefore, the function \( g(x) \) is one-to-one and has an inverse.

### Table (b) \( h(x) \)
| \( h(x) \) | \( x \)  |
|------------|----------|
| -2         | -12      |
| -1         | -9       |
| 0          | -6       |
| 1          | -3       |
| 2          | 0        |
| 3          | 3        |

- Each \( h(x) \) value is unique as well. Therefore, the function \( h(x) \) is also one-to-one and has an inverse.

### Table (c) \( j(x) \)
| \( j(x) \) | \( x \)  |
|------------|----------|
| -2         | 4        |
| -1         | 2        |
| 0          | 1        |
| 1          | 0.5      |
| 2          | 0.25     |
| 3          | 0.125    |

- Each \( j(x) \) value is unique. Hence, the function \( j(x) \) is one-to-one and has an inverse.

### Table (d) \( k(x) \)
| \( k(x) \) | \( x \)  |
|------------|----------|
| -2         | -5       |
| -1         | -3       |
| 0          | 1        |
| 1          | -1       |
| 2          | 3        |
| 3          | 3        |

- In this table, \( k(x) = 3 \) for both \( x = 2 \) and \( x = 3 \). Since two different inputs share the same output, this function is **not** one-to-one, and therefore it does not have an inverse.

### Summary:
- Functions \( g(x) \), \( h(x) \), and \( j(x) \) have inverses because they are one-to-one.
- Function \( k(x) \) does **not** have an inverse because it is not one-to-one (it repeats the value \( 3 \)).

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Would you like further explanation or have any specific questions?

Here are 5 follow-up questions to consider:
1. What is the definition of a one-to-one function?
2. How do we test if a function has an inverse from its graph?
3. What would be the inverse of the function \( g(x) \)?
4. Why can't a function that repeats outputs have an inverse?
5. How would the domain and range swap when finding the inverse of a function?

**Tip**: If a function is one-to-one, its graph will pass the horizontal line test: no horizontal line should intersect the graph more than once.

Determine which function(s) have an inverse function from the tables below. If an inverse function does not exist, provide a reason.

Learn how to determine if a function has an inverse by analyzing function tables. This problem involves understanding one-to-one functions, inverse functions, and applying the horizontal line test. Suitable for students in Grades 9-12.

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