Chapter 1
Numbers and Operations
Course: Gace Middle Grades Math 4 8 704
Subject: Gace Middle Grades Math 4 8 704
Chapter: Functions
1. Determines the domain and range of a function
The domain of a function is the complete set of x-values (inputs) for which the function produces a valid output. The range is the complete set of y-values (outputs) that the function can generate.
How to determine domain and range depends on how the function is presented:
From a table of values:
- The domain is the collection of all x-values listed in the table.
- The range is the collection of all y-values produced. Example: Table {(–2, 5), (0, 3), (3, 10)} Domain = {–2, 0, 3}, Range = {5, 3, 10}.
From an algebraic rule:
- Polynomial functions (e.g., y = x² – 3x + 2): domain is all real numbers, since polynomials are defined everywhere. Range depends on the shape of the graph (e.g., parabola has a minimum or maximum).
- Rational functions (e.g., f(x) = 1/(x – 4)): domain excludes any x-values that make the denominator zero. Here, x ≠ 4. Range excludes values that the function cannot output (often 0 for simple rational forms).
- Square root functions (e.g., f(x) = √(x – 2)): domain requires radicand ≥ 0. Here, x ≥ 2. Range is y ≥ 0 because square roots are nonnegative.
- Logarithmic functions (e.g., f(x) = log(x + 1)): domain requires argument > 0. Here, x > –1. Range is all real numbers.
- Exponential functions (e.g., f(x) = 2^x): domain is all real numbers. Range is y > 0, because exponential functions never produce negative outputs.