MTH 121 Chapter One: Introduction to Mappings and Functions of Real Variables

Flashcards covering definitions and application problems for mappings, functions, domains, and ranges based on MTH 121 lecture notes.

Function (Mapping)

A specific type of mapping where every input element from the domain corresponds to exactly one output value.

Mapping $x \rightarrow x$

A mapping where every element maps to itself, which defines a function.

Mapping $x \rightarrow \pm\sqrt{x}$

A mapping that does not define a function because it assigns two different values (both positive and negative) to a single input $x$.

Mapping $x \rightarrow x^2$

A mapping that defines a function where each input is associated with its square.

Domain

The set of all possible input values for which a function is defined; in the case of $f: A \rightarrow B$, it is set $A$.

Range

The set of all output values produced by a function; in the context of $f: A \rightarrow B, f(x)=2x$ for $A=\{1, 2, 3, 4\}$, it is the set of values obtained in $B$.

Ordered Pairs of $f(x)=2x$

The set of pairs representing the function given $A=\{1, 2, 3, 4\}$ and $B=\{2, 4, 6, 8\}$, expressed as $\{(1, 2), (2, 4), (3, 6), (4, 8)\}$.

Domain of $f(x) = \sqrt{x^2-5x+6}$

The set of all real numbers $x$ such that the expression under the square root is non-negative, specifically $x^2-5x+6 \ge 0$.

Domain of $g(x) = \frac{1}{x^2-5x+6}$

The set of all real numbers $x$ such that the denominator is not equal to zero, specifically $x^2-5x+6 \neq 0$.

Domain of $h(x) = \frac{x-1}{x+2}$

The set of all real numbers $x$ except for those that make the denominator zero, which is $x \neq -2$.