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Grade 9 · lesson 4 of 4

Choose a Branch before Substituting

A piecewise definition names both formulas and the conditions under which each applies.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For f(x)=x when 0≤x≤2, and f(x)=2+2(x−2) when x>2, find f(5). Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9.
  • Independent check inputs: For f(x)=x when 0≤x≤3, and f(x)=3+2(x−3) when x>3, find f(6). Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9.

Learn the idea

A line can be represented through slope and point, standard coefficients, or intercepts. Piecewise functions combine different rules on different domains, so choosing the correct branch is part of evaluation. A piecewise definition names both formulas and the conditions under which each applies. Compare the input with those conditions first, then evaluate only the selected formula.

Here is the method on the supplied teaching case. 5>2, so choose the second branch, not the first. Substitute: 2+2(5−2)=8. At the boundary the stated first branch applies.

Boundary symbols decide which branch applies at a breakpoint. Do not average branch formulas or choose the simpler one; evaluation must follow the explicitly supplied input intervals.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: For f(x)=x when 0≤x≤2, and f(x)=2+2(x−2) when x>2, find f(5). Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9. Solution: 5>2, so choose the second branch, not the first. Substitute: 2+2(5−2)=8. At the boundary the stated first branch applies. The final answer is 8. Boundary symbols decide which branch applies at a breakpoint. Do not average branch formulas or choose the simpler one; evaluation must follow the explicitly supplied input intervals.

Your turn

Check your reasoning

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Helpful tips
  • A piecewise definition names both formulas and the conditions under which each applies.
  • Boundary symbols decide which branch applies at a breakpoint.
For f(x)=x when 0≤x≤3, and f(x)=3+2(x−3) when x>3, find f(6).

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

9

6>3, so choose the second branch, not the first. Substitute: 3+2(6−3)=9. At the boundary the stated first branch applies.

Take it further

Changed-input task: For f(x)=x when 0≤x≤9, and f(x)=9+2(x−9) when x>9, find f(12). Candidate answers: (A) 14 (B) 15 (C) 16 (D) 30. Show the mathematical reason, not only the selected answer. Expected result: 15. 12>9, so choose the second branch, not the first. Substitute: 9+2(12−9)=15. At the boundary the stated first branch applies.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For f(x)=x when 0≤x≤4, and f(x)=4+2(x−4) when x>4, find f(7).
Need a hint?
  • A piecewise definition names both formulas and the conditions under which each applies.
  • Boundary symbols decide which branch applies at a breakpoint.
2. For f(x)=x when 0≤x≤5, and f(x)=5+2(x−5) when x>5, find f(8).
Need a hint?
  • A piecewise definition names both formulas and the conditions under which each applies.
  • Boundary symbols decide which branch applies at a breakpoint.
3. For f(x)=x when 0≤x≤6, and f(x)=6+2(x−6) when x>6, find f(9).
Need a hint?
  • A piecewise definition names both formulas and the conditions under which each applies.
  • Boundary symbols decide which branch applies at a breakpoint.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.