Functions, Domains, and Graphs: Foundations for Business Calculus

Interval Notation and Inequalities

Definition and Types of Intervals

Interval notation is a concise way to describe sets of real numbers, often used to specify domains of functions. Intervals can be finite or infinite, and may include or exclude endpoints depending on the type of inequality.

• Open Interval: $(a, b) = \{x \mid a < x < b\}$ (excludes endpoints)

• Closed Interval: $[a, b] = \{x \mid a \leq x \leq b\}$ (includes endpoints)

• Half-Open Intervals:

  • $[a, b) = \{x \mid a \leq x < b\}$ (includes left endpoint)

  • $(a, b] = \{x \mid a < x \leq b\}$ (includes right endpoint)

• Infinite Intervals:

  • $(b, \infty) = \{x \mid x > b\}$

  • $[b, \infty) = \{x \mid x \geq b\}$

  • $( -\infty, a ) = \{x \mid x < a\}$

  • $( -\infty, a ] = \{x \mid x \leq a\}$

• Entire Real Line: $( -\infty, \infty )$

Examples and Applications

• Domain of $f(x) = x^2$: $( -\infty, \infty )$

• Domain of $g(x) = \sqrt{x}$: $[0, \infty )$

• Domain of $h(x) = \frac{1}{x}$: $( -\infty, 0 ) \cup ( 0, \infty )$

• Domain of $f(x) = \sqrt{1 - x^2}$: $[ -1, 1 ]$

• Domain of $f(x) = \frac{1}{x - 1}$: $( -\infty, 1 ) \cup ( 1, \infty )$

• Domain of $g(x) = \frac{1}{\sqrt{x}}$: $( 0, \infty )$

Function Domains in Applications

Box Construction Problem

In applied calculus, domains are often restricted by physical constraints. Consider constructing a box from a 6 × 8 piece of cardboard by cutting equal squares from each corner and folding up the sides.

The process involves:

• Cutting squares of size x from each corner.

• Folding up the sides to form a box.

The dimensions of the resulting box are:

• Height: x

• Length: 8 - 2x

• Width: 6 - 2x

Volume Function

The volume as a function of x is:

• $V(x) = x(6 - 2x)(8 - 2x)$

• Expanded: $V(x) = 4x^3 - 28x^2 + 48x$

Domain of the Volume Function

Although $V(x)$ is defined for all real numbers, physical constraints restrict the domain:

• $x \geq 0$ (height cannot be negative)

• $6 - 2x \geq 0 \implies x \leq 3$

• $8 - 2x \geq 0 \implies x \leq 4$

• Thus, the domain is $[0, 3]$

Optimization: Calculus can be used to find the value of x that maximizes the volume, a classic optimization problem.

Review of Basic Functions and Graphs

Constant Functions

A constant function has the form $f(x) = k$, where $k$ is a fixed real number. Its graph is a horizontal line.

• Example: $y = 2$

Linear Functions

Linear functions are of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Graph: Straight line; slope $m$ determines direction (positive or negative).

• Y-intercept: $(0, b)$

Piecewise Defined Functions

Piecewise functions have different rules for different parts of their domain.

• Example: $f(x) = \begin{cases} 2x + 2 & x < 1 \\ -3x + 5 & x \geq 1 \end{cases}$

• Graph may have 'breaks' or 'jumps' at points where the rule changes.

• Open and closed circles indicate whether endpoints are included.

Absolute Value Function

The absolute value function $|x|$ is piecewise defined:

• $|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

• Graph is V-shaped, meeting at the origin.

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$, $a \neq 0$. Their graphs are parabolas.

• Orientation: Upward if $a > 0$, downward if $a < 0$.

• Axis of symmetry: $x = h$, where $h = -\frac{b}{2a}$

• Vertex: $(h, k)$, where $k = f(h)$

• Roots: Found using the quadratic formula:

  • $x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a }$

• Root classification:

  • No roots if $b^2 - 4ac < 0$

  • One (repeated) root if $b^2 - 4ac = 0$

  • Two distinct roots if $b^2 - 4ac > 0$

Polynomials in General

Polynomials of degree $n$ ($n > 2$) have more complex behavior than quadratics. Their general form is:

• $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• Roots are generally harder to find.

• Graphs may have multiple peaks and valleys (local maxima and minima).

• Calculus (tangent lines, derivatives) helps locate these features.

Summary Table: Interval Notation Types

Interval Notation	Inequality	Includes Endpoints?
(a, b)	a < x < b	No
[a, b]	a ≤ x ≤ b	Yes
[a, b)	a ≤ x < b	Left only
(a, b]	a < x ≤ b	Right only
(b, ∞)	x > b	No
[b, ∞)	x ≥ b	Left only
(−∞, a)	x < a	No
(−∞, a]	x ≤ a	Right only