Functions, Relations, Domain, and Range

Practice questions covering the definitions of functions, types of relations (one-to-one, many-to-one, one-to-many), domain and range identification, and the Vertical and Horizontal Line Tests.

What is the formal definition of a function according to the transcript?

A pairing of elements from one set $(X)$ to elements of another set $(Y)$, where every element in set $X$ is paired with exactly one element in set $Y$.

When is a relation $R$ from set $A$ to set $B$ considered one-to-one?

If each element in set $A$ is mapped only to one element in set $B$, and each element in set $B$ is mapped to only one element in set $A$.

What characterizes a many-to-one relation?

It is a relation where more than one element in set $A$ is mapped to the same one element in set $B$.

Which types of correspondence are classified as NOT functions?

one-to-many and many-to-many.

Define the 'domain' of a function $y=f(x)$.

The set of all possible $x$-values.

Define the 'range' of a function $y=f(x)$.

The set of all corresponding $y$-values.

What is the purpose of the Vertical Line Test?

It is used to determine whether a graph represents a function.

How do you determine if a graph is a function using the Vertical Line Test?

If a vertical line intersects the graph at only one point, it is a function; if it intersects at more than one point, it is not a function.

What is the purpose of the Horizontal Line Test?

It is used to determine whether a function represented by a graph is one-to-one.

In the context of coordinate pairs, what do the first and second coordinates represent?

The first coordinates represent the Domain, and the second coordinates represent the Range.

Is the set of ordered pairs $(1, 2), (1, 3), (1, 4), (1, 5), (1, 6)$ a function or not a function?

not function

Is the set of ordered pairs $(0, 0), (2, 2), (4, 4), (6, 6), (8, 8)$ a function or not a function?

function

Based on the 'Input and Output' activity, if a student earns $P80$ per hour, what is the salary for working 3 hours?

$160 + 80 = 240$ ($P240$).

According to the 'Domain or Range?' activity, what does the set $\{5, 10, 15, 20\}$ from the $y$-values of a graph represent?

Range $(R)$.

Identify the Domain for the mapping diagram: $10 \rightarrow 15$, $20 \rightarrow 25$, $30 \rightarrow 35$, $40 \rightarrow 45$.

$$\{10, 20, 30, 40\}$$

Why is the relation $\{(1,-5), (2,1), (3,0), (4,-5), (1,0)\}$ considered 'not a function'?

Because the element $1$ in the domain is paired with more than one element in the range ($-5$ and $0$).

As per the 'Fair Match Matters' activity, why can't one student have two different official ID numbers?

In a function-like system, every element (student) must be paired with exactly one unique identifier to maintain order and avoid confusion.