Lesson 3 of 6
Find real domains, describe ranges and read piecewise boundaries.
One idea at a time. Try the example, then keep going.
Big ideas · 1 of 12
Big ideas
Do this: Read one idea, then explain it in your own words.
The DomainAll the inputs you are allowed to use.Exactly: The set of input values for which a function is defined.Mix-up alert: A denominator cannot be zero; a real square root needs a nonnegative input.See this in Key words is the set of allowed inputs. For real-valued rules, never divide by zero or take an even root of a negative number.
The range contains outputs the rule actually produces on its stated domain. A formula alone does not mean every real output occurs.
A square bracket includes a finite endpoint; a parenthesis excludes it. Infinity is never an included endpoint, so use a parenthesis there.
A Piecewise functionA function that uses different rules for different inputs.Exactly: A function defined by different expressions on specified parts of its domain, assigning one output to each allowed input.Mix-up alert: Choose the rule by the input, not by the output.See this in Key words uses its input condition to select a rule. At a boundary, check the inequality sign before calculating.
Step-through
Do this: Step through the example. Predict the next step before revealing it.
Worked example: combine restrictions
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Do this: Cover the next step and try it yourself.
Input −4 works; input 1 does not.
Do this: Cover the next step and try it yourself.
Answer: 9.
Do this: Explain why the tempting shortcut fails.
Wrong idea: simplifying a formula makes every input allowed. Restrictions come from the original rule. For stays excluded even though the simplified expression is 1.
Big ideas
Do this: Read one idea and locate it on the example graph.
The DomainAll the inputs you are allowed to use.Exactly: The set of input values for which a function is defined.Mix-up alert: A denominator cannot be zero; a real square root needs a nonnegative input.See this in Key words records every x-coordinate reached by the graph. Separate pieces may leave gaps; do not fill those gaps just because both endpoints are visible.
The range records every y-coordinate reached anywhere on the graph. Combine the heights of all pieces, keeping a gap when no piece reaches those outputs.
A filled endpoint is included; an open circle is excluded. An open point does not by itself exclude its whole x-coordinate or y-coordinate: another filled point may use it.
Step-through
Do this: Sketch the stated segments, then follow the four steps beside the drawing.
The entire graph consists of two straight segments; no points join the gap.
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Do this: Cover each next step and collect the inputs and outputs yourself.
The missing inputs and missing outputs are different sets.
Worked example
Do this: Try this on paper before opening the worked answer. No online turn-in is needed.
Worked example
Do this: Predict the next step on paper, then compare the worked reasoning.
Combine them as .
Tip
Do this: Open your own class copy at the listed section.
Class handout reference · Unit 2, section 2.4, pages 9-14 and 24-30. Use your class copy.
Watch
Do this: Watch the public lesson, pause to try a step, then return to your practice.
The linked video is free to watch. The creator may offer optional paid resources; they are not needed for this lesson. If YouTube is blocked, use the lesson notes and worked examples here.
Ready when you are.
Activity 1 of 10 · 0 of 9 activities done
Optional packet practice
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Paper is for class. Try online at home if it helps. No points or turn-in record are created. Answers and working stay in this page’s memory and disappear when you leave or reload.
60 of 64 source prompts: 29 interval/inequality answer checks and 31 self-comparisons. On printed page 9, questions 1 to 5 check intervals only; draw their number lines on paper. Other graph drawings stay in the paper/PDF with source descriptions or coordinates here. Four graphs are omitted because their boundaries or continuation are ambiguous: PDF page 6 (printed 25), (h) and (i); PDF page 7 (printed 27), questions 7 and 8. Worked steps and hints are original instructional support.
Question 1 of 60
Put in interval notation and draw a graph of each inequality.
Online checks the interval only. Draw the requested number line on paper, then compare the drawing with the worked steps.
A bracket or filled point includes an endpoint; a parenthesis or open point excludes it. A ray toward infinity always uses a parenthesis at infinity.
Use when your teacher permits it. Keep working alongside it; closing the calculator keeps your calculations.
Flashcards
Do this: Say the answer and a reason before you flip.
Card 1 of 8
0 Know it0 Still learning
Think of the x-values you may use.
On the card: Enter or Space flips. Left and Right move. After revealing, 1 marks Still learning and 2 marks Know it. Tab reaches every control.
What is a domain?
The set of allowed inputs.
What is a range?
The outputs actually produced on the stated domain.
What restricts ?
x cannot equal −6.
What restricts over the real numbers?
.
Does include 2? Does it include 7?
It includes 2 but excludes 7.
What is the range of on all real inputs?
.
How do you evaluate a piecewise rule?
Choose the branch whose input condition is satisfied, then substitute.
What extra restriction can a model add?
Context can restrict inputs, such as whole nonnegative item counts.
True or false
Do this: Choose true or false; explain what would make a false statement correct.
Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Write two or three sentences, then compare with the model.
Start here if it helps: I would first identify...
Your answer is not saved anywhere. It disappears when you leave the page.
Self-test
Do this: Sketch each complete graph from its coordinates before choosing an answer.
4 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Sketch on paper, write your intervals here, then compare with the model.
Start here if it helps: My sketch has two separate pieces. The domain is...
Your answer is not saved anywhere. It disappears when you leave the page.
Write it
Do this: Write a mathematical argument, then compare its reasoning with the model.
Start here if it helps: My argument starts with...
Your answer is not saved anywhere. It disappears when you leave the page.
Match
Do this: Choose a word, then choose its meaning.
Connect the ideas
0 / 4 connectedChoose a term, then its meaning. Or drag its grip. Use Tab and Enter or Space; Escape clears.
Homework · 3 points
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Homework
Worth 3 points.
https://gradebridge-beryl.vercel.app/learn/precalculus/functions/domain-and-range#homework
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