3.2 Domain and Range

Functions are the backbone of algebra, describing relationships between variables. Domain and range define where these relationships exist and what values they produce. Understanding these concepts is crucial for solving real-world problems and mastering more advanced mathematical ideas.

In this section, we'll explore domain restrictions, piecewise functions, and practical applications. We'll also touch on advanced concepts like function composition and inverse functions. These ideas will help you analyze and manipulate functions with confidence.

Domain and Range of Functions

Domain restrictions on functions

• Domain is the set of all possible input values (usually $x$) for a function
  • Represented using set notation $\{x \mid x \in \mathbb{R}\}$ or interval notation $(-\infty, \infty)$
• Restrictions on the domain arise from:
  • Division by zero
    • Functions with denominators cannot have input values that make the denominator equal to zero
    • $f(x) = \frac{1}{x - 2}$ has a domain of $\{x \mid x \neq 2\}$ or $(-\infty, 2) \cup (2, \infty)$
  • Even roots (square root, fourth root, etc.)
    • Input values must result in a non-negative value under the even root
    • $f(x) = \sqrt{x + 1}$ has a domain of $\{x \mid x \geq -1\}$ or $[-1, \infty)$
  • Logarithms
    • Input values must be positive when using natural logarithm (ln) or common logarithm (log)
    • $f(x) = \ln(x - 3)$ has a domain of $\{x \mid x > 3\}$ or $(3, \infty)$
• Asymptotes can indicate domain restrictions in rational functions

Analysis of piecewise functions

• Piecewise functions consist of two or more sub-functions, each defined over a different part of the domain
  • Sub-functions usually denoted by $f_1(x)$, $f_2(x)$, etc.
• To find the domain of a piecewise function:
  1. Identify the domain of each sub-function
  2. Combine the sub-function domains, considering any overlaps or gaps
• Graphing piecewise functions:
  • Graph each sub-function within its respective domain
  • Use open or closed circles to indicate whether endpoints are included or excluded
  • $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases}$
    • $f_1(x) = x^2$ has a domain of $(-\infty, 1)$
    • $f_2(x) = 2x - 1$ has a domain of $[1, \infty)$
    • The domain of $f(x)$ is $(-\infty, \infty)$, with $f_1(x)$ and $f_2(x)$ meeting at $x = 1$
• Graphical representation helps visualize domain and range of piecewise functions

Real-world applications of domain and range

• Function notation: $f(x)$ represents the output value of the function for a given input value $x$
• Interval notation: Expresses a range of values using parentheses (exclusive) or brackets (inclusive)
  • $(2, 5]$ represents all values greater than 2 and less than or equal to 5
• Real-world context examples:
  • Car rental company charges a base fee of $50 plus$0.25 per mile driven
    • Domain: number of miles driven (non-negative real numbers) $[0, \infty)$
    • Range: total cost in dollars (values greater than or equal to the base fee) $[50, \infty)$
  • Rectangular garden has a perimeter of 60 feet
    • Domain: width of the garden (positive real numbers) $(0, 30)$
    • Range: length of the garden (positive real numbers) $(0, 30)$

Advanced Function Concepts

• Function composition: Combining two or more functions to create a new function
• Inverse function: A function that "undoes" the original function, swapping input and output
• One-to-one function: A function where each element of the codomain is paired with at most one element of the domain