Vocabulary Flashcards: Interval Notation, Field Axioms, and Functions

Vocabulary flashcards covering interval notation, field axioms, set theory, function transformations, iteration, and inverse functions based on lecture notes.

Increasing Function on an Interval

A function $y = f(x)$ defined on an interval $[a,b]$ is increasing on $[a,b]$ if whenever $x_1 < x_2$, then $f(x_1) < f(x_2)$.

Decreasing Function on an Interval

A function $y = f(x)$ defined on an interval $[a,b]$ is decreasing on $[a,b]$ if whenever $x_1 < x_2$, then $f(x_1) > f(x_2)$.

Even Number Axiom

Any number multiplied by $2$ is an even number, expressed as $x = 2m$ where $m$ is any integer.

Odd Number Axiom

Any even number plus or minus $1$ is an odd number, expressed as $y = 2n + 1$ where $n$ is any integer.

Number System Hierarchy

The containment structure where the set of natural numbers is contained in the set of integers, which is contained in rational numbers, contained in real numbers, and contained in complex numbers.

Most Restrictive Number System for $-7$

The set of integers ($\mathbb{Z}$), which is the most restrictive set to which $-7$ belongs.

Most Restrictive Number System for $\sqrt[4]{36}$

The set of rational numbers ($\mathbb{Q}$), because $\sqrt[4]{36} = \frac{2}{3}$.

Fixed Points of $F(x) = x^2$

The values of $x$ that remain invariant under the function, which for $F(x) = x^2$ are $x = 0$ and $x = 1$.

Fixed Point of $F(x) = \sqrt{x} + 2$

The fixed point value $x = 2$ where $F(x) = x$.

Horizontal Shift Transformation $f(x - 5)$

A function transformation that shifts the graph of the parent function $f(x)$ by $5$ units to the right.

Vertical Shift Transformation $f(x) - 5$

A function transformation that shifts the graph of the parent function $f(x)$ down by $5$ units.

Reflection over x-axis Transformation $-f(x)$

A function transformation that reflects the graph of $f(x)$ vertically over the x-axis.

Reflection over y-axis Transformation $f(-x)$

A function transformation that reflects the graph of $f(x)$ horizontally over the y-axis.

Vertical Compression Transformation $\frac{1}{2}f(x)$

A function transformation that compresses the graph of $f(x)$ vertically by a factor of $\frac{1}{2}$.

Inverse of $h(r) = -9r + 11$

The inverse relation $h^{-1}(r) = -\frac{1}{9}(r - 11)$, which is a valid function.

Inverse of $y = 5x^2 - 14$

The inverse relation given by $y = \pm \sqrt{\frac{1}{5}(x + 14)}$.