Mathematics 9: Relations and Functions

Practice questions covering the definitions of relations and functions, representations, mappings, and identifying domain and range from sets, graphs, and equations.

What is a relation $R$ from a non-empty set $A$ to a non-empty set $B$?

A subset of the Cartesian product set $A \times B$ derived by describing a relationship between the first and second elements of ordered pairs.

If $n(A) = p$ and $n(B) = q$, what is the formula for the total number of possible relations from set $A$ to set $B$?

$$2^{pq}$$

How is a function defined in relation to elements of sets $A$ and $B$?

A relation where every element of set $A$ has one and only one image in set $B$, meaning each first element is paired with exactly one second element.

In a function, what are the alternative names for the independent variable $x$ and the dependent variable $y$?

The variable $x$ is the argument and the variable $y$ is the image of the function.

What are three common ways that relations and functions can be expressed?

As a set of ordered pairs, a correspondence or mapping, or a graph.

Which types of correspondence are classified as functions?

One-to-one correspondence and many-to-one correspondence.

Why is a one-to-many correspondence not considered a function?

Because an $x$-value has two or more arrows branching to $y$-values, violating the rule that each input must have exactly one output.

What is the Vertical Line Test?

A test used on a graph where a relation is a function if no two points lie on the same vertical line.

Define 'Domain' and 'Range'.

Domain is the set of all inputs or first elements; Range is the set of all outputs or second elements actually used.

What does the use of square brackets in interval notation, such as $[-1, 5]$, indicate?

It indicates that the endpoints (in this case $-1$ and $5$) are included in the interval.

What is the rule when writing the domain and range for a set of values with repeating numbers?

Do not repeat values; each value should be written only once.

What are the domain and range of the linear equation $y = x + 2$?

Domain: $\{x | x \text{ is an element of } \text{R}\}$ or $(-\infty, \infty)$; Range: $\{y | y \text{ is an element of } \text{R}\}$ or $(-\infty, \infty)$.

What is the range of the quadratic equation $y = x^2$ and why?

Range: $[0, \infty)$ because the square of any number is always zero or positive, making negative values impossible for $y$.

What is the domain restriction for the rational equation $y = \frac{1}{x}$?

$\{x | x \text{ is an element of } \text{R}, x \neq 0\}$ because zero cannot be used as a divisor.

What is the rule for the radicand when finding the domain of a relation involving a radical with an even index?

The radicand must be nonnegative, meaning it must be greater than or equal to zero.

What is the restriction for the denominator when finding the domain and range of a fraction?

The denominator must not be equal to zero.