Relations and Functions in Business Calculus

Part I: Relations and Functions

Definition of Relations and Functions

A relation is a rule of correspondence between two sets, typically denoted as D (domain) and R (range), and is expressed as a set of all ordered pairs (x, y) where x is in D and y is in R. The domain is the set of all possible input values (x), and the range is the set of all possible output values (y).

• Function: A relation is a function if for each x in D there is only one y in R. The notation is y = f(x).

• Independent variable: The variable x (input).

• Dependent variable: The variable y (output), or f(x).

• If the domain is not specified, it is assumed to be the largest set of real numbers for which the function is defined.

• Range: The set of all values y = f(x) as x varies over the domain D.

Example: Determine whether a relation is a function and give the domain and range.

• Relation: {(1,1), (2,2), (3,3), ...} is a function. Domain: {1,2,3,...}, Range: {1,2,3,...}

• Relation: {(5,5), (2,-3), (5,-5)} is not a function (since x=5 is paired with two different y values).

Functions Defined by Equations

If a relation is given by an equation in x and y, it represents a function if for each x there is only one real solution for y.

• Example: x2 + y2 = 4 is not a function (for some x, there are two y values).

• Example: 2x + 3y = 4 is a function (for each x, there is only one y).

Evaluating Functions

To evaluate a function, substitute the given value into the function's formula.

• Example: If $f(x) = \frac{1}{x^2 - 4}$, then:

  • $f(4) = \frac{1}{4^2 - 4} = \frac{1}{12}$

  • $f(2)$ is undefined since $x=2$ is not in the domain (division by zero).

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and is defined as:

$\frac{f(x+h) - f(x)}{h}$, where $h \neq 0$

• Example: For $f(x) = 3x - 2x^2$:

  • $f(x+h) = 3(x+h) - 2(x+h)^2 = 3x + 3h - 2(x^2 + 2xh + h^2) = 3x + 3h - 2x^2 - 4xh - 2h^2$

  • $f(x+h) - f(x) = (3x + 3h - 2x^2 - 4xh - 2h^2) - (3x - 2x^2) = 3h - 4xh - 2h^2$

  • Difference quotient: $\frac{3h - 4xh - 2h^2}{h} = 3 - 4x - 2h$

Domain of a Function

The domain of a function is the set of all real numbers for which the function is defined.

• Example: $f(x) = \frac{3x}{3x+5}$ Domain: $x \neq -\frac{5}{3}$

• Example: $g(x) = \frac{\sqrt{x-1}}{x^2-4}$ Domain: $x \geq 1$ and $x \neq 2, x \neq -2$

Graphs of Relations and Functions

The graph of a relation in x and y is the set of all points (x, y) in the xy-plane that correspond to the ordered pairs of the relation.

• If a relation is a function, then for every x in the domain there is only one y in the range.

• The vertical line test: A set of points (x, y) in the xy-plane is the graph of a function if and only if each vertical line intersects the graph at most once.

Even and Odd Functions

Functions can be classified as even, odd, or neither, based on their symmetry properties.

• Even function: $f(-x) = f(x)$ for all x in the domain. The graph is symmetric with respect to the y-axis.

• Odd function: $f(-x) = -f(x)$ for all x in the domain. The graph is symmetric with respect to the origin.

• Examples:

  • $f(x) = x^2$ is even.

  • $f(x) = x^3$ is odd.

  • $f(x) = 2x + 1$ is neither even nor odd.

  • $f(x) = 2$ is even.

  • $f(x) = x^2 - 1$ is even.

Arithmetic of Even and Odd Functions

The sum or difference of even and odd functions follows these rules:

Operation	Result
Even + Even	Even
Odd + Odd	Odd
Even + Odd	Neither
Even - Even	Even
Odd - Odd	Odd
Even - Odd	Neither

Applications of Functions

Cost Function Example

Suppose the cost C(x) of renting a car for x days (or fractions of a day) includes a fixed drop-off charge and a daily rate. For example, if the drop-off charge is $44 and the daily rate is $54, then:

$C(x) = 44 + 54x$

• Example calculations:

  • $C(2/3) = 44 + 54 \times \frac{2}{3} = 44 + 36 = 80$

  • $C(1) = 44 + 54 \times 1 = 98$

  • $C(1.4) = 44 + 54 \times 1.4 = 119.6$

• Is C(x) linear? Yes, because it has the form $C(x) = mx + b$.

• Independent variable: x (number of days or fractions of a day)

• Dependent variable: C (cost of renting the car)

Area as a Function of Width

Given a rectangular field with a fixed perimeter, the area can be expressed as a function of the width.

• Let perimeter $P = 2(w + l) = 6000$

• Then $l = 3000 - w$

• Area $A(w) = w \times l = w(3000 - w)$

• Domain: $0 < w < 3000$ (since width and length must be positive)

• Graph: The area function is a downward-opening parabola, increasing for $0 < w < 1500$ and decreasing for $1500 < w < 3000$.

Interpretation: The area increases as width increases up to a maximum at $w = 1500$, then decreases as width increases further.