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Worked solution

A missing output between two pieces

Given that f(x)={x20≤x<2,7−x2≤x<3.f(x)=\begin{cases}x^2&0\le x<2,\\7-x&2\le x<3.\end{cases} Sketch the graph of y=f(x)y=f(x) and state the range of the function.

2010/CJC/MYE/12(a)(i)

CJC · Functions · Example 11

Sketch the two pieces

Answer

Closed (0,0)(0,0) and (2,5)(2,5); open (2,4)(2,4) and (3,4)(3,4).

Draw y=x2y=x^2 only on [0,2)[0,2), starting at the included origin and approaching the excluded (2,4)(2,4). Draw y=7−xy=7-x only on [2,3)[2,3), from included (2,5)(2,5) to excluded (3,4)(3,4). Do not join the jump at x=2x=2; its value is 5, not 4.

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Quadratic from closed (0,0) to open (2,4); line from closed (2,5) to open (3,4). Horizontal y=4 is excluded by both pieces.
Quadratic from closed (0,0) to open (2,4); line from closed (2,5) to open (3,4). Horizontal y=4 is excluded by both pieces.
Quadratic from closed (0,0) to open (2,4); line from closed (2,5) to open (3,4). Horizontal y=4 is excluded by both pieces.
How did this part go?

State the range

Answer

Rf=[0,4)∪(4,5]R_f=[0,4)\cup(4,5].

The quadratic piece reaches every output in [0,4)[0,4); the decreasing line reaches every output in (4,5](4,5]. Neither reaches 4. Union these output sets without filling the missing value. The source range is visibly completed only in the Teacher copy; native Student extraction contains answer text hidden from the rendered answer space.

How did this part go?

Use the printed calculator procedure

Answer

Use the screen to inspect shape; determine exact endpoints algebraically.

  1. Press Y=, then MATH, scroll to B:piecewise(, press ENTER, set Pieces:2, select OK, then ENTER. The template contains two rule/condition rows. 2. Enter X2X^2 with 0≤X<20\le X<2, and 7−X7-X with 2≤X<32\le X<3. For conditions use 2nd → MATH → CONDITIONS → 7:≤X and X<. 3. Press GRAPH. 4. Use ZOOM → 1:ZBox, box the desired region, then ENTER. ZBox magnifies the selected portion. The screen may not resolve endpoints: calculate their coordinates and inclusion algebraically. Menu labels can vary by calculator model; the reconstruction illustrates the source sequence rather than emulating a device.

Enter and inspect a piecewise function

  1. 1. Create two pieces

    Press Y=, then MATH. Select B:piecewise( and press ENTER. Set Pieces:2, select OK, then press ENTER.

  2. 2. Pair each rule with its condition

    Enter the rules in the two rows of Y1. For the interval template use 2nd → MATH → CONDITIONS → 7:≤X and X<.

    First rule
    X², 0≤X and X<2
    Second rule
    7−X, 2≤X and X<3

  3. 3. Inspect the branches

    Press GRAPH. If the short branches are hard to distinguish, use ZOOM → 1:ZBox, box the region of interest, then press ENTER. This magnifies the selected portion.

  4. 4. Check endpoints from the conditions

    Determine inclusion from the inequalities. At x=2 the first rule is excluded and the second gives f(2)=5. Do not join the jump or fill the missing output 4.

    Check endpoints from the conditions
    EndpointIncluded?
    (0, 0)Yes
    (2, 4)No
    (2, 5)Yes
    (3, 4)No

    Range: [0, 4) ∪ (4, 5]

Screen sampling may hide open endpoints. Calculate their coordinates and inclusion algebraically. Menu labels can vary by model; the retained screens illustrate the source sequence rather than emulate a device.

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Four source stages: select MATH B:piecewise with 2pieces; enter X² with 0≤X<2 and 7−X with 2≤X<3 using CONDITIONS 7; GRAPH; ZOOM 1:ZBox and ENTER. Find exact endpoints algebraically.
Four source stages: select MATH B:piecewise with 2pieces; enter X² with 0≤X<2 and 7−X with 2≤X<3 using CONDITIONS 7; GRAPH; ZOOM 1:ZBox and ENTER. Find exact endpoints algebraically.
Four source stages: select MATH B:piecewise with 2pieces; enter X² with 0≤X<2 and 7−X with 2≤X<3 using CONDITIONS 7; GRAPH; ZOOM 1:ZBox and ENTER. Find exact endpoints algebraically.
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