Business Calculus: Functions, Graphs, and Models – Comprehensive Study Notes

Functions, Graphs, and Models

Introduction to Functions

Functions are foundational in calculus and describe relationships between variables. Understanding their properties and graphs is essential for modeling business scenarios.

• Definition: A function is a rule that assigns to each input exactly one output.

• Notation: $f(x)$ denotes a function named $f$ with input $x$.

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Example: $f(x) = 2x + 3$ is a linear function with domain $(- , )$ and range $(- , )$.

Types of Functions and Their Graphs

Common function types include linear, quadratic, cubic, absolute value, and rational functions. Each has a characteristic graph and algebraic form.

• Linear Functions: $f(x) = mx + b$

  • Graph: Straight line with slope $m$ and $y$-intercept $b$.

• Quadratic Functions: $f(x) = ax^2 + bx + c$

  • Graph: Parabola opening up if $a > 0$, down if $a < 0$.

• Cubic Functions: $f(x) = ax^3 + bx^2 + cx + d$

  • Graph: S-shaped curve.

• Absolute Value Functions: $f(x) = |x|$

  • Graph: V-shaped, vertex at the origin.

• Rational Functions: $f(x) = \frac{p(x)}{q(x)}$

  • Graph: May have vertical/horizontal asymptotes.

Graphing Basic Functions

Graphing involves plotting points and understanding transformations such as shifts, stretches, and reflections.

• Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts the graph right by $h$ units.

• Reflection: $-f(x)$ reflects across the $x$-axis.

• Example: $f(x) = (x - 2)^2 + 3$ is a parabola shifted right 2 units and up 3 units.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} x + 2 & x < 0 \\ x^2 & x 0 \end{cases}$

• Graph each piece on its respective interval.

Intercepts and Symmetry

Finding intercepts and analyzing symmetry helps in sketching graphs.

• $x$-intercept: Set $f(x) = 0$ and solve for $x$.

• $y$-intercept: Evaluate $f(0)$.

• Even Function: $f(-x) = f(x)$ (symmetric about $y$-axis).

• Odd Function: $f(-x) = -f(x)$ (symmetric about origin).

Intervals and Domain

Understanding intervals is crucial for defining domains and ranges.

• Interval Notation:

  • $(a, b)$: All $x$ such that $a < x < b$ (open interval).

  • $[a, b]$: All $x$ such that $a x b$ (closed interval).

  • $(a, b]$: $a < x b$ (half-open interval).

• Example: The domain of $f(x) = \sqrt{x}$ is $[0, )$.

Vertical and Horizontal Asymptotes

Asymptotes describe the behavior of functions as $x$ approaches certain values.

• Vertical Asymptote: Occurs where the function grows without bound as $x$ approaches a value (e.g., denominator zero in rational functions).

• Horizontal Asymptote: Describes the end behavior as $x o$ or $x o -$.

• Example: $f(x) = \frac{1}{x}$ has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.

Summary Table: Common Functions and Their Properties

Function	Equation	Graph Shape	Domain	Range
Linear	$f(x) = mx + b$	Straight line	$(- , )$	$(- , )$
Quadratic	$f(x) = ax^2 + bx + c$	Parabola	$(- , )$	$[k, )$ or $(- , k]$
Absolute Value	$f(x) = |x|$	V-shape	$(- , )$	$[0, )$
Rational	$f(x) = \frac{1}{x}$	Hyperbola	$x 0$	$(- , 0) (0, )$

Applications in Business

Functions model cost, revenue, and profit in business. For example, a linear cost function $C(x) = mx + b$ models total cost as a function of units produced.

• Example: If $C(x) = 5x + 200$, then the fixed cost is $200$and the variable cost per unit is $5$.

Additional info:

• Some graphs and examples were inferred from standard calculus curriculum and the visible sketches in the notes.

• Piecewise and rational function examples were expanded for clarity.