To determine if $f$ is a function, we need to verify whether each input $x$ is paired with exactly one output $y$. A function has the defining property that no $x$-value is associated with more than one $y$-value.

Analyzing the relation:

The given relation is:
$\{(1,2), (4,2), (3,7), (-1,8)\}.$

Step 1: Check uniqueness of $x$-values

The $x$-values are: $\{1, 4, 3, -1\}.$ Each $x$-value is unique, meaning no $x$-value is repeated. Therefore, $f$ satisfies the condition of being a function regarding $x$-values.

Step 2: Check uniqueness of $y$-values

The $y$-values are: $\{2, 2, 7, 8\}.$ The $y$-value $2$ is repeated. However, this does not violate the definition of a function because a function allows multiple $x$-values to map to the same $y$-value.

Conclusion:

$f$ is a function because the $x$-values are unique. The correct statement is:

"F is a function because the $x$-values are unique."

Would you like further clarification on this? Let me know!

Here are 5 related questions to explore:

1. How do you determine if a relation is a function?
2. Can a function have repeated $y$-values? Why or why not?
3. What is the difference between one-to-one functions and general functions?
4. How do you check if a function is one-to-one using the horizontal line test?
5. How can the concept of functions apply to real-world scenarios?

Tip: Remember, a function only fails if a single $x$-value maps to more than one $y$-value. Repeated $y$-values do not disqualify it as a function.To determine whether $f$ is a function, we need to check the definition of a function: a relation $f$ is a function if each $x$-value corresponds to exactly one $y$-value.

Analyzing the relation $f$:

1. The given pairs are: $(1, 2), (4, 2), (3, 7), (-1, 8)$
2. The $x$-values are: $1, 4, 3, -1$.
3. The $y$-values are: $2, 2, 7, 8$.

Steps to determine if $f$ is a function:

• Check the uniqueness of $x$-values: Each $x$-value is unique: $1, 4, 3, -1$. This satisfies the condition for $f$ to be a function.

• Check the $y$-values (optional for functions): While $y$-values $2, 2, 7, 8$ are not all unique, this does not violate the definition of a function. $y$-values can repeat in a function as long as $x$-values do not map to multiple $y$-values.

Correct Answer:

F is a function because the $x$-values are unique.

Let me know if you’d like further clarification! Here are some related questions to explore:

1. How does the uniqueness of $x$-values define a function?
2. Can a function have multiple $y$-values that are the same?
3. What happens if an $x$-value maps to more than one $y$-value?
4. What is the difference between relations and functions?
5. Can you identify whether an equation is a function using graphs?

Tip: Always focus on the $x$-values when deciding if a relation is a function. Repeated $y$-values do not disqualify a function.