Multiple Choice

1. In a mathematical relation represented by the ordered pairs $(3, 7)$, $(5, 9)$, and $(8, 7)$, which set of numbers represents the DOMAIN of the relation?

   $\{7, 9\}$

   The second numbers in the pairs: $\{7, 9, 7\}$

   All numbers listed: $\{3, 5, 7, 8, 9\}$

   The first numbers (the inputs): $\{3, 5, 8\}$

2. What is the most accurate description of a mathematical relation?

   A set of ordered pairs that connects inputs to outputs.

   A rule that always uses addition or subtraction.

   A relationship where every input has exactly one output.

   A picture showing shapes connected by lines.

3. The special rule for a function is that every INPUT must be linked to only one OUTPUT. Which set of ordered pairs is NOT a function because it breaks this rule?

   $(10, 1)$, $(11, 2)$, $(12, 3)$

   $(1, 4)$, $(2, 5)$, $(3, 4)$

   $(2, 8)$, $(2, 9)$, $(3, 10)$

   $(5, 5)$, $(6, 6)$, $(7, 7)$

4. If a relation has the ordered pairs $(1, 3)$, $(2, 3)$, $(3, 3)$, and $(4, 3)$, what is the RANGE of this relation?

   All numbers used: $\{1, 2, 3, 4, 3\}$

   $\{3\}$

   The domain is $\{3\}$.

   $\{1, 2, 3, 4\}$

5. All functions are relations, but not all relations are functions. Why is this true?

   Relations must always follow a mathematical formula.

   Relations are much easier to draw than functions.

   Functions only use positive numbers, but relations can use negative numbers.

   A function is a special type of relation that follows the extra rule of having only one output per input.

6. Which set of ordered pairs *must* be a function?

   $(4, 1)$, $(4, 2)$, $(4, 3)$

   $(5, 8)$, $(6, 8)$, $(5, 9)$

   $(1, 10)$, $(2, 10)$, $(1, 11)$

   $(1, 1)$, $(2, 2)$, $(3, 3)$, $(4, 4)$

7. The Vertical Line Test (VLT) tells us if a graph is a function. If a vertical line crosses the graph in TWO different places, what does that immediately tell you?

   One input ($x$) is matched to two different outputs ($y$).

   The relation is definitely a function.

   The range numbers are repeating.

   The graph must be perfectly straight.

8. Imagine the graph of a perfect circle. If you used the Vertical Line Test on the circle, would it pass or fail, and what does the result mean?

   Fail, because a vertical line hits the circle twice, meaning it is a relation, not a function.

   Pass, because a circle is a continuous shape.

   Fail, because the inputs (domain) are restricted.

   Pass, meaning every input has only one output.

9. In a relation, if the ordered pairs include $(-5, 12)$ and $(-5, 15)$, we know it is NOT a function. Why?

   The input $-5$ is paired with two separate outputs, violating the unique output rule.

   Because the outputs 12 and 15 are too far apart.

   Because the output numbers are different.

   Because the input number is negative.

10. A graph PASSES the Vertical Line Test if:

    Any vertical line you draw touches the graph at most one time.

    The graph goes up from left to right.

    The graph only uses positive numbers.

    The vertical line never touches the graph.

11. A mapping diagram shows: Input $A \rightarrow 5$ and Input $A \rightarrow 9$. Is this a function?

    No, because Input A has two different outputs (5 and 9).

    Yes, because 5 and 9 are different outputs.

    No, because it only has one input.

    Yes, because it uses letters and numbers.

12. Which description of a graph would guarantee that the relation is NOT a function?

    A graph that connects $(-1, 4)$ and $(5, 4)$.

    A graph that forms a straight line going diagonally.

    A graph that forms a perfect, horizontal line.

    A graph that includes the points $(2, 1)$ and $(2, 7)$.

13. Why does the Vertical Line Test work to identify functions?

    Because only non-functions have graphs that look like vertical lines.

    A vertical line means all the points on that line share the exact same input ($x$ value).

    Because vertical lines are parallel to the x-axis (inputs).

    Because it helps us count how many points are in the domain.

14. A store owner tracks sales where the Input is (Item Sold) and the Output is (Price Paid). Which set represents the DOMAIN in this scenario?

    The total sales amount for the day.

    The money paid by the customer.

    The list of items sold.

    The size or color of the items.

15. Three of these relations are functions. Which one is NOT a function?

    $\{(-1, 1), (0, 0), (1, 1)\}$

    $\{(5, 5), (6, 6), (7, 7)\}$

    $\{(3, 4), (5, 6), (3, 7)\}$

    $\{(10, 5), (20, 10), (30, 15)\}$

16. Which geometric shape, when graphed centered at the origin, would FAIL the Vertical Line Test the most dramatically?

    A parabola that opens to the right (like $x = y^2$).

    A square.

    A straight diagonal line.

    A perfectly horizontal line.

17. If we are checking if a relation is a function, which set must we ensure does NOT have any repeating values linked to different partners?

    The set of ordered pairs.

    The set of all numbers in the relation.

    The set of outputs (the range).

    The set of inputs (the domain).

18. Why is the rule "one input, one unique output" so important for defining a function?

    It means the graph will always be a straight line.

    It makes sure the domain and range have the same number of elements.

    It prevents the outputs from being confusingly close together.

    It guarantees that for any specific input value, there is only one predictable result or value.

19. A relation $R$ is defined by the set of ordered pairs $\{(a, b), (c, d), (e, f)\}$. Which mathematical expression correctly represents the domain of $R$?

    $\{a, c, e\}$

    $\{y | (x, y) \in R\}$

    $\{x | x \text{ is the second element in an ordered pair in } R\}$

    $\{(a, c, e) | (a, b) \in R \text{ and } (c, d) \in R \text{ and } (e, f) \in R\}$

20. The strict definition distinguishing a function from a general relation mandates that for every input, there must be one unique output. If a relation's graph fails the Vertical Line Test (VLT), what is the direct algebraic violation of this mandate?

    The range of the relation is larger than the domain.

    The graph must be disconnected (discontinuous) at the point of intersection.

    At least one $x$-value corresponds to two or more distinct $y$-values, meaning the relation contains pairs $(a, b)$ and $(a, c)$ where $b \neq c$.

    The relation contains ordered pairs where the $y$-values are identical for different $x$-values, such as $(a, c)$ and $(b, c)$ where $a \neq b$.

21. Consider the relation $S = \{(4, -1), (k, 7), (-2, 3), (4, k)\}$. For this relation $S$ to be classified as a function, which specific condition must the variable $k$ satisfy?

    $k$ must not be equal to $-1$, $7$, or $3$.

    $k$ must be a negative number.

    $k$ must be greater than or equal to $3$.

    $k$ must be equal to $-1$.

22. The equation $x = y^2 + 1$ defines a relation. If this relation were graphed, why would the Vertical Line Test (VLT) confirm that it is NOT a function?

    For any $x > 1$, solving the equation for $y$ yields $y = \pm \sqrt{x-1}$, resulting in two distinct $y$-values for a single input $x$.

    The domain of the relation is restricted to $x \geq 1$.

    The graph opens upward, meaning it only passes the Horizontal Line Test.

    For every positive $x$-value, the graph would have an $x$-intercept at $y=0$.

23. A student graphs a relation and observes that a vertical line at $x=a$ intersects the graph exactly three times. Which of the following must be true about the set of ordered pairs defining this relation?

    The relation contains three distinct ordered pairs that share the first element $a$: $(a, y_1), (a, y_2), \text{ and } (a, y_3)$.

    The relation must contain three different $x$-coordinates that map to the same $y$-coordinate.

    The relation passes the VLT for all $x$-values except $x=a$.

    The domain of the relation is restricted to $\{x | x \neq a\}$.

24. Given four different sets of ordered pairs, which set represents a function?

    $Q = \{ (5, 1), (1, 5), (-5, 1), (1, -5) \}$

    $P = \{ (5, 5), (5, 5), (-5, 5), (5, -5) \}$

    $M = \{ (10, 2), (10, -2), (2, 10), (-2, 10) \}$

    $N = \{ (1, 1), (-1, 1), (1, 0), (-1, 0) \}$

25. If a relation $G$ is known to be a function, which of the following statements comparing its domain $D$ and its range $R$ must be logically true?

    The number of elements in $D$ must be greater than or equal to the number of elements in $R$, or $|D| \geq |R|$.

    Every element in $R$ must be mapped to by exactly one element in $D$.

    The sets $D$ and $R$ must be identical ($D = R$).

    The set $D$ must contain more elements than the set $R$, or $|D| > |R|$.

26. Consider a graph that is defined by the piecewise relation: $y = x^2$ for $x \leq 0$ and $y = \sqrt{x}$ for $x > 0$. If the function requirement is strictly observed, which modification to the definition is necessary to ensure the entire graph passes the VLT?

    The first definition must be changed to $y = -x^2$ for $x \leq 0$.

    The two pieces of the graph must be connected at the origin $(0, 0)$.

    The domain $x > 0$ must be changed to $x < 0$.

    No modification is necessary, as the input $x=0$ is only used once in the definition ($y=0^2 = 0$), and all other $x$ inputs are unique to their definition.

27. Why is the concept of a function, defined by the "one unique output" rule, particularly important in differential calculus?

    It ensures that the input and output values always remain non-negative.

    It guarantees that if the function is smooth, the slope (derivative) at any given point $x$ must be uniquely defined and not ambiguous.

    It guarantees that the function's domain and range are symmetric.

    It allows for the unique mapping of all real inputs to real outputs.

28. A student claims that a relation shown in a table must be a function if all the $y$-values in the table are unique. Which set of ordered pairs below provides a counterexample to this flawed logic?

    $\{ (2, 10), (3, 11), (4, 12), (5, 10) \}$

    $\{ (-1, 1), (-2, 4), (-3, 9), (-4, 16) \}$

    $\{ (5, 1), (5, 2), (6, 3), (7, 4) \}$

    $\{ (1, 5), (2, 6), (3, 7), (4, 8) \}$