Business Calculus: Functions, Domains, Interval Notation, and Piecewise Functions

Functions and Their Domains

Understanding Functions

A function is a rule that assigns each input value (from the domain) to exactly one output value. In business calculus, functions often represent relationships such as cost, revenue, or profit as a function of quantity produced or sold.

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

• Example: $f(x) = \frac{5x}{3x-1}$

Finding the Domain of a Function

The domain of a function is the set of all input values ($x$) for which the function is defined.

• Restrictions:

  • The denominator cannot be zero (for rational functions).

  • No imaginary numbers (inputs must yield real outputs).

  • For logarithmic functions, the argument must be positive.

• Example: For $f(x) = \frac{5x}{3x-1}$, set $3x-1 \neq 0 \Rightarrow x \neq \frac{1}{3}$.

• Domain in interval notation: $(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Interval Notation

Interval notation is a concise way to describe sets of numbers, often used for domains and ranges.

• Closed interval: $[a, b]$ includes both endpoints $a$ and $b$.

• Open interval: $(a, b)$ excludes both endpoints.

• Half-open intervals: $[a, b)$ or $(a, b]$ include only one endpoint.

• Infinite intervals: $(-\infty, a)$, $(a, \infty)$, etc.

• Example: $x \in [2, 5)$ means $2 \leq x < 5$.

Piecewise Functions

Definition and Examples

A piecewise function is defined by different expressions depending on the input value.

• Notation: $f(x) = \begin{cases} x^2 & x < 2 \\ 2x+1 & x \geq 2 \end{cases}$

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

• Evaluate $f(1)$: $1^2 = 1$ (since $1 < 2$)

• Evaluate $f(3)$: $2 \times 3 + 1 = 7$ (since $3 \geq 2$)

Rational Functions

Definition

A rational function is a function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Domain: All real numbers except where $Q(x) = 0$.

• Example: $f(x) = \frac{x^2+2x+1}{x^2-1}$

• Find where denominator is zero: $x^2-1=0 \Rightarrow x=1$ or $x=-1$

• Domain: $(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$

Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth: If $a > 1$, the function models exponential growth.

• Decay: If $0 < a < 1$, the function models exponential decay.

• Example: $f(x) = 2^x$

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, written as $f(x) = \log_a(x)$.

• Domain: $x > 0$

• Range: $(-\infty, \infty)$

• Example: $f(x) = \log_{10}(x)$

Point-Slope Form of a Line

Equation and Application

The point-slope form of a line is useful for writing the equation of a line given a point and a slope.

• Formula: $y - y_1 = m(x - x_1)$

• Example: Through $(2, 5)$ with slope $m = 4$: $y - 5 = 4(x - 2)$

Additional info:

• Some content inferred from context and standard business calculus curriculum, such as the importance of domain restrictions for rational and logarithmic functions.

• Piecewise and rational functions are commonly used in modeling business scenarios with different rules or constraints.