Business Calculus Study Notes: Nonlinear Functions, Domain & Range, Rational Functions, Asymptotes, and Linear Systems

Nonlinear Functions and Function Basics

Function Notation and Evaluation

A function is a mathematical relationship where each input (usually x) has exactly one output (usually y). The notation f(x) denotes a function of x.

• Example: If f(x) = 3 + x, then f(2) = 3 + 2 = 5.

• Revenue Function: R(x) = p \times x, where p is price per unit and x is the number of units.

• Break-Even Quantity: The x-value where revenue equals cost, R(x) = C(x).

• Profit Function: P(x) = R(x) - C(x).

Vertical Line Test

The vertical line test is a visual method to determine if a graph represents a function. If any vertical line crosses the graph more than once, it is not a function.

• Valid Function: A vertical line intersects the graph at only one point for each x-value.

• Invalid Function: A vertical line intersects the graph at more than one point for some x-values.

Domain and Range of Functions

Domain

The domain of a function is the complete set of possible values for the independent variable (x) that produce real outputs.

• The denominator of a fraction cannot be zero.

• The value under a square root must be positive (for real numbers).

• To find the domain, identify all x-values that are allowed in the function.

Range

The range of a function is the set of all possible output values (y) after substituting the domain values.

• Find the minimum and maximum y-values by substituting x-values.

• Check if y is always positive, negative, or restricted from certain values.

• Sketching the graph can help visualize the range.

Interval Notation for Range

• [a, b]: Both endpoints a and b are included.

• (a, b): Endpoints a and b are not included.

• {c}: Only the value c is included.

Rational Functions and Asymptotes

Rational Functions

A rational function is a function defined by the ratio of two polynomials, typically written as $f(x) = \frac{P(x)}{Q(x)}$.

Asymptotes

An asymptote is a line that a graph approaches but never touches. There are three main types: vertical, horizontal, and oblique.

• Vertical Asymptote: Occurs where the denominator of a rational function equals zero.

• Horizontal Asymptote: Determined by comparing the degrees of the numerator and denominator.

• Oblique Asymptote: Occurs when the degree of the numerator is one higher than the denominator.

Finding Horizontal Asymptotes

To determine the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator:

If...	Then...
The degree of the numerator < the degree of the denominator	There is a horizontal asymptote at y = 0
The degree of the numerator = the degree of the denominator	There is a horizontal asymptote at y = ratio of leading coefficients
The degree of the numerator > the degree of the denominator	There is no horizontal asymptote

Linear Functions and Systems

Linear Functions

A linear function is an equation that represents a straight line, typically written as $y = mx + b$.

• Exponent, Power, Degree: These terms refer to the same concept; for example, in $x^4$, the exponent, power, and degree are 4.

Linear Systems

A linear system is a collection of linear equations. Solving a linear system means finding values for the variables that satisfy all equations simultaneously.

• Example: Solve the system:

  • $3x + 10y = 115$

  • $11x + 4y = 95$

Types of Solutions for Two Equations with Two Unknowns

There are three possible types of solutions for a system of two linear equations:

• Unique Solution: The two graphs intersect at a single point. The system is consistent and the solution is given by the coordinates of the intersection.

• Inconsistent: The graphs are distinct parallel lines; there is no solution common to both equations.

• Dependent: The graphs are the same line; any solution of one equation is also a solution of the other, resulting in infinitely many solutions.

Applications: Area Function and Domain

Rectangular Lot with Fencing

Consider a rectangular lot where a fence is to be built against a brick wall, forming three sides of the rectangle. The contractor uses a fixed amount of fencing, and the length of the wall is l, width is w.

• Equation for Area: The area A as a function of length l is $A(l) = l \times w$.

• Constraint: The total fencing used is $l + 2w = 200$ (since only three sides are fenced).

• Domain: The domain consists of all positive values of l such that w is also positive and the total fencing does not exceed 200 m.

Summary Table: Horizontal Asymptotes

If...	Then...
The degree of the numerator < the degree of the denominator	There is a horizontal asymptote at y = 0
The degree of the numerator = the degree of the denominator	There is a horizontal asymptote at y = ratio of leading coefficients
The degree of the numerator > the degree of the denominator	There is no horizontal asymptote

Recap

• Use the vertical line test to determine if a graph is a function.

• Domain: x-values (avoid zero in denominator and negative under square root).

• Range: y-values (find minimum and maximum y).

• Linear systems can have unique, inconsistent, or dependent solutions.