How do you determine the domain and range of a linear function, including reasonable continuous or discrete values for a real-world situation?
Determine the domain and range of a linear function in mathematical problems, and reasonable domain and range values (continuous and discrete) for real-world situations, representing them using inequalities (TEKS A.2A).
A STAAR Algebra I answer on the domain and range of linear functions (TEKS A.2A), continuous versus discrete situations, reasonable real-world values, and representing domain and range with inequalities.
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What this topic is asking
TEKS A.2A asks you to find the domain and range of a linear function, both for a bare mathematical function and for a real-world situation where only certain inputs and outputs make sense. The key distinction the standard tests is continuous versus discrete, and you must express the domain and range using inequalities. This sits in Reporting Category 3 and connects to interpreting any function model.
Domain and range of a bare linear function
For a function with no context, picture the graph. A non-vertical line stretches infinitely left and right, so every is an input: the domain is all real numbers. It also stretches infinitely up and down (as long as the slope is nonzero), so every is an output: the range is all real numbers.
The exception is a horizontal line , whose range is the single value , and a vertical line , which is not a function at all.
Continuous versus discrete in context
In a real-world problem, the variable's meaning limits the sensible values.
- Continuous quantities can take any value in an interval: time, distance, weight, temperature. Their domain or range is an unbroken interval like .
- Discrete quantities take only separate values, usually whole numbers: people, tickets, cars, laps. Their domain or range is a set of distinct points, described as whole numbers within a range.
The deciding question is "can this quantity be split into fractions sensibly?" You can drive 3.5 miles (continuous), but you cannot sell 3.5 tickets (discrete).
Representing with inequalities
The standard requires inequality notation. A domain "from 0 to 20" is written ; a range "at least 0" is . For a discrete situation, the inequality still bounds the values, with the understanding that only whole numbers inside the bound are used.
How STAAR examines this topic
- Multiple choice. Choose the domain or range, with continuous-versus-discrete as the discriminator.
- Inline choice. Pick the domain and range descriptions from dropdowns, including "all real numbers" for an unrestricted function.
- Drag and drop. Match a situation to its reasonable domain or range.
A clarifying idea is that context shrinks the domain: the bare function has domain all reals, but a printer that runs for at most 8 minutes restricts the reasonable domain to . The standard is really asking which values make sense in the situation, and the matching range is then found by evaluating the function at the endpoints of that restricted domain.
Try this
Q1. State the domain and range of with no restriction. [1 point]
- Cue. Domain all real numbers; range all real numbers.
Q2. A team buys $7 shirts, up to 30 of them. Reasonable domain? [1 point]
- Cue. Discrete, (whole numbers).
Exam-style practice questions
Practice questions written in the style of TEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
STAAR (style)1 marksMultiple choice. A taxi charges a flat 2 per mile, modeled by , for whole numbers of miles up to 20. Which best describes the domain? (A) Discrete, (B) Continuous, (C) Discrete, all real numbers (D) Continuous, Show worked answer →
The correct answer is (A).
The domain is the set of inputs, here the number of miles . The problem restricts to whole numbers from 0 to 20, so the domain is discrete (separate values, not a continuous range) with . Choice (B) treats miles as continuous, but "whole numbers of miles" makes it discrete; choice (D) confuses the domain with a cost value.
STAAR (style)2 marksInline choice. For the function with no restriction, the domain is [all real numbers / x at least 0 / 0 to 10] and the range is [all real numbers / y at least negative 1 / 0 to 10].Show worked answer →
The domain is all real numbers and the range is all real numbers.
A non-vertical line with no contextual restriction extends infinitely in both directions, so every real number is a possible input (domain) and every real number is a possible output (range). The restricted options describe contextual situations, not the bare function. Only horizontal lines have a single-value range; has slope 3, so its outputs cover all reals.
Related dot points
- Solve linear equations in one variable, including those requiring the distributive property and those with variables on both sides, and identify equations with one solution, no solution, or infinitely many (TEKS A.5A).
A STAAR Algebra I answer on solving one-variable linear equations (TEKS A.5A), the inverse-operations routine, the distributive property, variables on both sides, and recognizing one solution, no solution, or infinitely many.
- Write linear functions that model the relationship between two quantities from a description, table, or graph, write an equation representing a functional relationship, and evaluate functions in function notation (TEKS A.2C, A.2G, A.12B).
A STAAR Algebra I answer on writing linear functions to model situations, identifying initial value and rate, function notation f(x), and evaluating functions (TEKS A.2C, A.2G, A.12B).
- Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems (TEKS A.3A).
A STAAR Algebra I answer on graphing linear functions and reading their key features (TEKS A.3A) - the x-intercept, y-intercept, zeros, and slope - from slope-intercept and standard form, including the hot-spot graphing item type.
- Calculate the rate of change (slope) of a linear function represented tabularly, graphically, or algebraically, and interpret slope and intercepts as rate and initial value in context (TEKS A.3A, A.3B).
A STAAR Algebra I answer on finding slope and rate of change from tables, graphs, two points, and contexts (TEKS A.3A, A.3B), the slope formula on the reference sheet, and interpreting slope and intercepts in real-world situations.
- Determine the domain and range of quadratic functions and represent them using inequalities, and describe representations of quadratic functions in relation to their solutions and the real-world situations they model (TEKS A.6A, A.6B).
A STAAR Algebra I answer on the domain and range of quadratic functions (TEKS A.6A, A.6B), why the range is bounded by the vertex, representing with inequalities, and connecting representations to real-world models.
Sources & how we know this
- STAAR Algebra I Assessed Curriculum — Texas Education Agency (2024)
- 19 TAC Chapter 111, Algebra I (TEKS), Adopted 2012 — Texas Education Agency (2012)