College Algebra Chapter 3

Relation

Any set of ordered pairs, $(x, y)$.

Function

A relation that maps each input in the domain to exactly one output in the range (informally, no $x$-coordinates repeat).

Independent Variable

The input variable of a function, represented by $x$.

Dependent Variable

The output variable of a function, represented by $y$.

One-to-One Function

A relation where each input has exactly one output and each output has exactly one input (informally, no $x$-coordinates repeat and no $y$-coordinates repeat).

Vertical Line Test

A visual test used to determine whether a graph represents a function.

Horizontal Line Test

A visual test used to determine whether the graph of a function represents a one-to-one function.

Domain

A set containing all of the inputs, or $x$-coordinates, of a function.

Range

A set containing all of the outputs, or $y$-coordinates, of a function.

Set-Builder Notation

A symbolic notation such as $\{x \mid x \le 5\}$ (read as "the set of $x$-values such that $x$ is less than or equal to $5$") used to specify domain or range.

Interval Notation

A notation that describes a continuous interval of real numbers using parentheses and brackets, such as $(-\infty, 5]$ for all numbers less than or equal to $5$.

Infinity ($\infty$)

A mathematical concept that is not a number and therefore can never be included using square brackets in interval notation.

Domain of Polynomials

Unless stated otherwise, the domain for all polynomial functions is $(-\infty, \infty)$.

Parent Function: (Increasing) Line

The linear parent function $y = x$ (or $y = 1x + 0$) with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$.

Parent Function: Parabola

The quadratic parent function $y = x^2$ with domain $(-\infty, \infty)$ and range $[0, \infty)$, centered at $(0, 0)$.

Parent Function: Cubic Function

The parent function $y = x^3$ with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$.

Parent Function: Square Root Function

The parent function $y = \sqrt{x}$ with domain $[0, \infty)$ and range $[0, \infty)$.

Parent Function: Hyperbola

The rational parent function $y = \frac{1}{x}$ with domain $(-\infty, 0) \cup (0, \infty)$ and range $(-\infty, 0) \cup (0, \infty)$, which does not touch either axis.

Parent Function: Inverse-Square Function

The parent function $y = \frac{1}{x^2}$ with domain $(-\infty, 0) \cup (0, \infty)$ and range $(0, \infty)$, which does not touch either axis.

Standard Equation of a Circle

The relation $(x-h)^2 + (y-k)^2 = r^2$ with center $(h, k)$ and radius $r$, which is not a function.

Parent Function: Semicircle (Centered at Origin)

The function $y = \sqrt{r^2 - x^2}$ with domain $[-r, r]$ and range $[0, r]$.

Average Rate of Change

The slope of the line segment connecting two points across a closed interval $[a, b]$, calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Constant Function Interval

An interval on a graph where a function is neither increasing nor decreasing.

Extrema

The extreme values of a function, representing the points with the largest or smallest $y$-coordinates on a curve (plural form of extremum).

Maximum

The point(s) on a curve with the largest value for a $y$-coordinate.

Minimum

The point(s) on a curve with the smallest value for a $y$-coordinate.

Relative Maximum

A local maximum point occurring where a function changes from increasing to decreasing, forming the highest point on a smooth rounded "hill" within an open interval.

Relative Minimum

A local minimum point occurring where a function changes from decreasing to increasing, forming the lowest point in a "valley" within an open interval.

Absolute Maximum

A global maximum point that is the highest point on the entire curve.

Absolute Minimum

A global minimum point that is the lowest point on the entire curve.

Pascal's Triangle

A triangular array of numbers whose rows provide the numerical coefficients used to expand any binomial expression, such as $(x + y)^n$.

Dividing by $h$

An algebraic reduction step in calculating average rates of change that is mathematically equivalent to multiplying by the reciprocal, $\frac{1}{h}$.

Composite Function

A function formed by evaluating one function inside another, denoted as $(f \circ g)(x) = f(g(x))$, where the order of evaluation matters.

Function Multiplication vs. Function Composition

Two distinct operations on functions: multiplication evaluates the product of their outputs, $(f \cdot g)(x) = f(x)g(x)$, which is not equivalent to the composition $(f \circ g)(x) = f(g(x))$.

Composition of Functions

The process of combining functions so that the output of one function becomes the input of another, denoted by $(f \circ g)(x) = f(g(x))$.

Conditions for Adding and Subtracting Functions

The requirements that inputs must share the same units (or have no units) and that outputs must be numbers so algebraic operations can be applied.

Domain of a Composite Function

The set of inputs $x$ in the domain of $g$ for which the value $g(x)$ is contained in the domain of $f$.

Informal Rule for Composite Domain

The procedure of identifying a composite function's domain by excluding values that make the innermost function undefined, as well as any values that make the overall composite function undefined.

Function Composition Modeling: $g(M(h))$

The composite representation determining gallons of gas used after driving $h$ hours, where $M(h)$ outputs miles from hours and $g(n)$ uses those miles as an input to output gallons of gas.

Target Output Modeling: $g(M(h)) = 8$

The equation set up to determine the specific driving time in hours ($h$) required for a vehicle to consume exactly $8$ gallons of gas.

Standard Transformation Form

The equation structure $\frac{y - k}{b} = f\left(\frac{x - h}{a}\right)$, where $a$ controls horizontal stretching/shrinking, $h$ controls horizontal translation, $b$ controls vertical stretching/shrinking, and $k$ controls vertical translation.

Vertical Stretch

A transformation of the form $g(x) = b \cdot f(x)$ where multiplying the output by a constant (such as $b = 2$) multiplies the $y$-coordinates and doubles their distance from the $x$-axis.

Horizontal Compression

A transformation of the form $h(x) = f(c \cdot x)$ with $c > 1$ (such as $f(3x)$), which corresponds to $a = \frac{1}{3}$ and divides each $x$-coordinate by $3$.

Horizontal Translation

A shift of a parent function along the $x$-axis represented by $f(x - h)$, shifting left when $h < 0$ (such as $s(x) = f(x + 4)$ where $h = -4$) and right when $h > 0$.

Vertical Translation

A shift of a parent function along the $y$-axis represented by $r(x) = f(x) + k$, shifting upward when $k > 0$ (such as $k = 5$) and downward when $k < 0$.

Factoring Horizontal Arguments

The algebraic technique of factoring out the coefficient of $x$ within an inner expression, such as rewriting $2x - 10$ as $2(x - 5)$, to correctly isolate the horizontal shift ($h = 5$) and scale factor ($a = \frac{1}{2}$).