TEKSForge
Lesson library
A.1(A)·Algebra II·Foundations for FunctionsReadiness

Domain and Range

Identify domains and ranges of functions and find corresponding values from rules and graphs.

Students connect domain inputs to range outputs by evaluating rules and reading the horizontal and vertical extent of graphs.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Algebra II released practice50 minutes

Learn the concept

Inputs and outputs

The domain consists of allowed inputs, usually x-values. The range consists of actual outputs, usually y-values. A point (a,b)(a,b) means that input a corresponds to output b.

Evaluate the whole rule

To find f(a)f(a), replace every x with a, simplify the entire numerator, and then divide. Finding an output is not the same as solving for an input.

Read the extent of a graph

Scan left to right for the domain and bottom to top for the range. A closed endpoint is included; an open endpoint is excluded unless another point supplies that coordinate. An open right endpoint can give an upper bound without a greatest domain element.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 47 · 1 point

Two segments on a grid from negative 7 to 7. One runs from a closed dot at (negative 5, 7) down to an open dot at (0, negative 3). The other runs from a closed dot at (0, 4) down to a closed dot at (6, 2).
Piecewise function graphOriginal visual from the released STAAR test.

Worked example

For g(x)=3x+75g(x)=\frac{3x+7}{5}, find the output for input 4. Then find the domain of the line segment with closed endpoints (−6,2)(-6,2) and (9,7)(9,7).

  1. Substitute: g(4)=3(4)+75g(4)=\frac{3(4)+7}{5}.
  2. Simplify: g(4)=195=3.8g(4)=\frac{19}{5}=3.8.
  3. The segment covers all horizontal coordinates from −6-6 to 9 and includes both endpoints.

The output is 3.8; the segment has domain −6≤x≤9-6\le x\le9 and greatest domain value 9.

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Identifying Parent Functions

UT IPSI · Kid2Kid / Texas Education Agency

Review graph features and input-output relationships before reading domain and range.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. Find h(−2)h(-2) if h(x)=4x−16h(x)=\frac{4x-1}{6}.

    Show answer and reasoning

    −1.5-1.5

    h(−2)=−8−16=−96=−1.5h(-2)=\frac{-8-1}{6}=-\frac96=-1.5.

  2. 2. A line segment runs from an open point (1,8)(1,8) to a closed point (7,−4)(7,-4). State its domain and range.

    Show answer and reasoning

    Domain: 1<x≤71<x\le7; range: −4≤y<8-4\le y<8.

    Project the segment onto each axis. The closed point includes 7 and −4-4; the open point excludes 1 and 8.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 24 · 1 point

If 5 is an element in the domain of f(x)=7x−224f(x) = \frac{7x - 22}{4}, what is the corresponding element in the range? Record your answer.

Show STAAR answer and explanation

3.25

Evaluate the function at the domain value: f(5)=7(5)−224=134=3.25f(5)=\frac{7(5)-22}{4}=\frac{13}{4}=3.25.

13: 13 is the numerator; divide by 4 to get the range value 3.25.

Ask the tutor about this question

Archived released STAAR Algebra II EOC · 2013 · Item 47 · 1 point

Two segments on a grid from negative 7 to 7. One runs from a closed dot at (negative 5, 7) down to an open dot at (0, negative 3). The other runs from a closed dot at (0, 4) down to a closed dot at (6, 2).
Piecewise function graphOriginal visual from the released STAAR test.

What is the greatest value in the domain of the function graphed on the grid?

  1. 2
  2. 6
  3. 4
  4. 7
Show STAAR answer and explanation

6

The domain is the set of x-values. The graph runs from the closed dot at (−5,7)(-5,7) to the closed dot at (6,2)(6,2), so the greatest x-value is 6.

2: 2 is the y-value of the right endpoint, which belongs to the range.

4: 4 is a y-value, where the right segment meets the y-axis.

7: 7 is the greatest y-value, part of the range rather than the domain.

Ask the tutor about this question
Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Distinguish domain values from range values.
  • Evaluate a function at a specified input.
  • Read graph endpoints to identify domain limits.

Instructional sequence

  1. 1

    Engage

    5 min

    Sort ordered-pair coordinates into inputs and outputs.

  2. 2

    Direct instruction

    10 min

    Model substitution using parentheses and the order of operations.

  3. 3

    Guided practice

    13 min

    Evaluate rational linear rules and compare input-output pairs.

  4. 4

    Compare and verify

    15 min

    Project graph points onto each axis and discuss endpoint inclusion.

  5. 5

    Closure

    7 min

    Explain the greatest domain value and complete the exit ticket.

Misconceptions to address

“The highest point gives the greatest domain value.”

Teacher move: Domain uses horizontal coordinates; locate the farthest-right point.

“The numerator is the final output.”

Teacher move: A fraction bar groups the numerator; finish the division.