courses › Algebra

Composition, inverses, domains, and piecewise rules.

Pen and paper is fine · no calculator needed why?

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Opens at level 27.

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Builds on: Lines & slope (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Piecewise functions par 18 s not tried yet

  worked example

  f(x) = 2x + 8 when x ≤ 0, and x² when x > 0. What is f(0) + f(5)?

  Answer: 33

  1. f(0) uses the first rule (x ≤ 0): 2 × 0 + 8 = 0 + 8 = 8.
  2. f(5) uses the second rule (x > 0): 5² = 25.
  3. Total: 8 + 25 = 33.

• Composition par 20 s not tried yet

  worked example

  f(x) = 2x + 6 and g(x) = x − 4. What is g(f(2))?

  Answer: 6

  1. Work inside out: f(2) = 2 × 2 + 6 = 4 + 6 = 10.
  2. Then g(10) = 10 − 4 = 6.

• Inverse value par 20 s not tried yet

  worked example

  f(x) = 3x − 10. What is f⁻¹(−31)?

  Answer: -7

  1. f⁻¹(−31) is the input that gives −31: solve 3x − 10 = −31.
  2. 3x = −21, so x = −7.

• Domain par 16 s not tried yet

  worked example

  What is the domain of f(x) = √(−2 − x)?

  1. x ≤ 2
  2. all real x
  3. x < −2
  4. x ≥ −2
  5. x ≤ −2

  Answer: x ≤ −2

  1. A square root needs a non-negative inside: −2 − x ≥ 0, so x ≤ −2.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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