Linear Functions and Equations in Business Calculus

Notion of a Function and Relation

Definition and Properties

Understanding functions and relations is foundational in business calculus, as they describe how variables interact in mathematical models.

• Relation: Any set of ordered pairs $(x, y)$. The set of all $x$-values is called the domain, and the set of all $y$-values is called the range.

• Function: A relation where each element in the domain is assigned to exactly one element in the range.

Example: For the relation $ig\{(1, 1), (2, 4), (3, 9), (4, 16), (5, 25)ig\\}$:

• Domain: $ig\{1, 2, 3, 4, 5\big\}$

• Range: $ig\{1, 4, 9, 16, 25\big\}$

Testing for Functions

• Each $x$-value must be paired with only one $y$-value for a relation to be a function.

• Equations like $x + y^2 = 3$ are not functions because a single $x$ can correspond to multiple $y$ values.

Example:

• For $y = f(x)$, $x$ is the independent variable, $y$ is the dependent variable.

• Given $f(x) = 2x^2 - x + 1$, calculate $f(1)$, $f(-2)$, $f(a)$, $f(a+h)$:

  • $f(1) = 2(1)^2 - 1 + 1 = 2$

  • $f(-2) = 2(-2)^2 - (-2) + 1 = 8 + 2 + 1 = 11$

  • $f(a) = 2a^2 - a + 1$

  • $f(a+h) = 2(a+h)^2 - (a+h) + 1$

Linear Functions

Definition and Graph

A linear function is defined by an equation of the form:

$f(x) = ax + b$

• $a$ and $b$ are real number constants.

• The graph of $y = ax + b$ is a straight line.

Coordinate System and Quadrants

• The x-axis is horizontal; the y-axis is vertical. They intersect at the origin $(0, 0)$.

• Any point in the plane is represented as $(x, y)$.

Quadrant	Condition
I	$x > 0$, $y > 0$
II	$x < 0$, $y > 0$
III	$x < 0$, $y < 0$
IV	$x > 0$, $y < 0$

Slope of a Line

Definition and Calculation

The slope of a line measures its steepness and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points.

$\text{Slope} = \frac{\text{Rise}}{\text{Run}}$

• Given two points $A(x_1, y_1)$ and $B(x_2, y_2)$, the slope $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• The slope is independent of which two points are chosen on the line.

• The slope of a vertical line is not defined.

Types of Slope

Type	Description
Positive Slope	Line rises from left to right
Negative Slope	Line falls from left to right
Zero Slope	Horizontal line
No Slope	Vertical line (undefined)

Example Calculation

• Find the slope of the line passing through $P(2, 1)$ and $Q(8, 5)$:

$m = \frac{5 - 1}{8 - 2} = \frac{4}{6} = \frac{2}{3}$

This means for every 3 units moved to the right, the line rises 2 units.

Graphing Linear Equations

General Approach

• To graph a linear equation, plot at least two points that satisfy the equation and draw a straight line through them.

• Intercepts are often used:

  • x-intercept: Set $y = 0$ and solve for $x$.

  • y-intercept: Set $x = 0$ and solve for $y$.

Linear Models for Depreciation

Application in Business

Linear functions are used to model depreciation, where the value of an asset decreases at a constant rate over time.

• General form: $V(t) = mt + b$

• $t$ is time, $V(t)$ is value, $m$ is the rate of depreciation (negative), $b$ is the initial value.

Example:

• Machine bought for $8000, value after 10 years is $500.

• Find $m$: $m = \frac{500 - 8000}{10 - 0} = \frac{-7500}{10} = -750$

• Depreciation function: $V(t) = -750t + 8000$

• Value after 7 years: $V(7) = -750 \times 7 + 8000 = -5250 + 8000 = 2750$

Rate of Change

Definition and Importance

• The rate of change of a linear function is its slope.

• For linear functions, the rate of change is constant.

• In business, rate of change can represent cost per unit, revenue per item, or depreciation per year.

Solving Equations with One Variable

Techniques and Properties

• To solve an equation, isolate the variable on one side using equivalent transformations.

• Properties of equality:

  • Addition: If $A = B$, then $A + C = B + C$

  • Multiplication: If $A = B$, then $AC = BC$

• Linear equations have the variable to the first power and graph as straight lines.

Example:

• Solve $7x - 4 = 3x + 8$:

$7x - 4 = 3x + 8$ $7x - 3x = 8 + 4$ $4x = 12$ $x = 3$

Summary Table: Types of Equations

Type	Example	Graph
Linear	$y = 2x + 3$	Straight line
Nonlinear	$y = x^2$	Curve (parabola)

Additional info:

• These notes are based on slides for "Applied Mathematics for Business QMS110" and are directly relevant to Business Calculus.

• All examples and definitions are expanded for clarity and completeness.