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) $g(x)$

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

Table (b) $h(x)$

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

Table (c) $j(x)$

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

Table (d) $k(x)$

• 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 $3$).

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.