Lesson 4 of 4
Select branches by input and handle graph endpoints explicitly.
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Big ideas · 1 of 8
Big ideas
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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 defines different formulas on stated input intervals. Select the condition first and evaluate only that branch.
A closed graph dot includes an endpoint; an open dot excludes it. Write matching or for a closed endpoint and or for an open one.
The total domain is the union of branch input sets. The total range is the union of branch outputs, calculated using each branch's own input restriction.
Find a line equation for each segment or ray, then attach its input condition. At shared boundaries, ensure each allowed input has exactly one output.
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Worked example: choose the endpoint
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The second branch gives outputs , which already includes 3, so the full range is .
Do this: Explain why the tempting shortcut fails.
Wrong idea: both formulas must be used at a boundary. Their input conditions decide which formula applies. If two included rules assign different outputs at one input, the proposed relation is not a function.
Worked example
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Worked example
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The right expression includes 2 and gives 4, so .
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Flashcards
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The inequality tells you the branch.
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How do you evaluate a piecewise rule?
Choose the input condition first, then calculate with its formula.
What does an open dot on a graph mean?
That branch excludes the endpoint.
What does a closed dot mean?
That branch includes the endpoint.
Can a piecewise function jump?
Yes; each allowed input still needs only one output.
How do you find the total domain?
Take the union of allowed input sets from all branches.
How do you find the total range?
Find outputs on each branch's restricted inputs, then take their union.
What must a graph description provide for a line piece?
Enough point or slope information plus its input interval and endpoint inclusion.
Can two branches include the same input with different outputs?
No; that would violate the function property.
True or false
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Self-test
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Self-test
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Write it
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Homework · 3 points
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Homework
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https://gradebridge-beryl.vercel.app/learn/precalculus/linear-functions/piecewise-functions#homework
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