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Algebra 2 9-12 / Week 08 / Monday
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Week 08 · The Function Toolkit

Monday

Building a function
// Build, move, compare, cross
⏱ about 20 min

Monday: Building a Function

The chalkboard wall at the back of the Glass House is finally clean, and Comet has the first stick of chalk.

"A square bed with side x has area x²," she says. "Four walkway strips along it, each x long and one wide: 4x."

"So the whole plot is x² + 4x," Wren says. "You added two rules. What do you notice?"

"That it is a new rule," Comet says. "Bed plus walkway, one function."

Wren tries another. "Rows on a bench: x + 3. Seedlings per row: 4x. Multiply them for the count."

Nova projects a tray with a rim around it. "Would you like a hint? Some rules feed into others."

"The tray side with its rim is x + 2. Its area is that squared," Comet says. "A rule inside a rule."

The chalkboard wall of overlapping curves in the Glass House; Wren points where two cross while Comet and Nova watch.

Three ways to build a function

  • Add or subtract: (f + g)(x) = f(x) + g(x). The plot: x² + 4x = x² + 4x.
  • Multiply: (f × g)(x) = f(x) × g(x). The bench: (x + 3)(4x) = 4x² + 12x.
  • Compose: f(g(x)) means do g first, then f. The tray: A(r(x)) = (x + 2)² = x² + 4x + 4.

A solved problem to study

  1. The tray area rule is A(s) = s², and the side with its rim is r(x) = x + 2.
  2. The crew wants the area of a tray with its rim when the inner side is 3.
  3. Work inside out. First r(3) = 5.
  4. Then A(5) = 5² = 25.
  5. Check with the expanded rule: x² + 4x + 4 at x = 3 is 25. It matches.
  6. Written as one function: A(r(x)) = (x + 2)². The rim rule sits inside the area rule.

Order matters in a composition. r(A(3)) would square first, then add 2: a different number.

Every length here is the crew's own tape-measure reading on the wall, not a fact about real trays.

The wall rules this week

Glass House ruleBuilt howRule
Bed plus walkwaysx² + 4xx² + 4x
Seedlings on the bench(x + 3) × (4x)4x² + 12x
Tray with rimA(r(x)) with r(x) = x + 2x² + 4x + 4
BUILD AND EVALUATE
  • Read the question.
  • Tap your answer.
Bed f(x) = x², walkways g(x) = 4x. What is the whole plot (f + g)(5)?
Rows r(x) = x + 3, seedlings per row p(x) = 4x. How many seedlings is (r × p)(2)?
Area A(s) = s², rim rule r(x) = x + 2. What is A(r(3))?
Now the other order: r(x) = x + 2 and A(s) = s². What is r(A(3))?
The plot rule is x² + 4x. What is the plot area when x = 4? Type the number.
WHY THIS EXERCISEAn added function is evaluated piece by piece and then summed. The new rule gives the same total.
StatementTrue or false?
(f + g)(x) means add the two outputs at the same x.?
f(g(x)) means do f first, then g.?
(3 + 2) × (3 + 2) = 25?
(x + 3)(4x) expands to 4x² + 12x.?
4 × 4 + 4 × 4 = 32?
WHY THIS EXERCISECombining functions is algebra you already know, wearing function notation.
Try it
Write two rules on index cards: p(x) = x + 1 and q(x) = 3x. Find p(q(2)) and q(p(2)).
Were they the same? Write one sentence about why the order changed the answer.
Draw the square bed with its four walkway strips. Label the bed x², each strip x, and the whole plot x² + 4x.

Good building. Tomorrow the parabola on the wall slides, stretches and flips.