Lesson Worksheet: Step Functions Mathematics

In this worksheet, we will practice identifying and analyzing a step function’s domain, range, intercepts, end behavior, continuity, and decreasing and increasing intervals.

Q1:

State the domain and range of the step function whose graph is shown.

  • ADomain: โˆ’2โ‰ค๐‘ฅ<4, range: {โˆ’8,โˆ’2,6.5,4}
  • BDomain: โˆ’8<๐‘ฅโ‰ค6.5, range: {โˆ’8,โˆ’2,6.5}
  • CDomain: โˆ’2โ‰ค๐‘ฅ<4, range: {โˆ’2,0.5,4}
  • DDomain: โˆ’8โ‰ค๐‘ฅโ‰ค6.5, range: {โˆ’2,4}
  • EDomain: โˆ’8<๐‘ฅโ‰ค6.5, range: {โˆ’2,0.5,4}

Q2:

State the domain and range of the following function: ๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉโˆ’2,โˆ’12<๐‘ฅโ‰คโˆ’1112,โˆ’1<๐‘ฅ<10,1โ‰ค๐‘ฅโ‰ค4.

  • ADomain: โˆ’12<๐‘ฅโ‰ค4, range: ๏ฌโˆ’12,โˆ’1,1,112,4๏ธ
  • BDomain: โˆ’2<๐‘ฅโ‰ค112, range: {โˆ’12,โˆ’1,1,4}
  • CDomain: โˆ’2โ‰ค๐‘ฅโ‰ค112, range: {โˆ’12,โˆ’1,1,4}
  • DDomain: โˆ’12<๐‘ฅโ‰ค4, range: ๏ฌโˆ’2,112,0๏ธ
  • EDomain: โˆ’12โ‰ค๐‘ฅโ‰ค4, range: ๏ฌโˆ’2,112,0๏ธ

Q3:

Evaluate the following function, ๐‘“, for the given values of ๐‘ฅ.

๐‘“(โˆ’3)

๐‘“(2)

๐‘“(0)

Q4:

Find the rule of the following function.

  • A๐‘“(๐‘ฅ)=๏ฒโˆ’1,โˆ’5โ‰ค๐‘ฅ<โˆ’23,โˆ’2โ‰ค๐‘ฅ<15,1โ‰ค๐‘ฅโ‰ค5.
  • B๐‘“(๐‘ฅ)=๏ฒโˆ’1,โˆ’5โ‰ค๐‘ฅ<โˆ’23,โˆ’2โ‰ค๐‘ฅโ‰ค15,1โ‰ค๐‘ฅโ‰ค5.
  • C๐‘“(๐‘ฅ)=๏ฒโˆ’1,โˆ’5โ‰ค๐‘ฅโ‰คโˆ’23,โˆ’2<๐‘ฅโ‰ค15,1โ‰ค๐‘ฅโ‰ค5.
  • D๐‘“(๐‘ฅ)=๏ฒโˆ’1,โˆ’5โ‰ค๐‘ฅโ‰คโˆ’23,โˆ’2โ‰ค๐‘ฅโ‰ค15,1โ‰ค๐‘ฅโ‰ค5.
  • E๐‘“(๐‘ฅ)=๏ฒโˆ’1,โˆ’5<๐‘ฅ<โˆ’23,โˆ’2<๐‘ฅ<15,1<๐‘ฅ<5.

Q5:

Which of the following functions is a step function?

  • A
  • B
  • C
  • D
  • E

Q6:

State the domain and range of the function ๐‘“(๐‘ฅ)=๏ฒโˆ’3,โˆ’5<๐‘ฅโ‰ค1,4,1<๐‘ฅ<4,6,4โ‰ค๐‘ฅ<8.

  • ADomain: [โˆ’3,6), range: {โˆ’3,4,8}
  • BDomain: (โˆ’5,8], range: {โˆ’3,4,6}
  • CDomain: [โˆ’5,8), range: {โˆ’3,1,6}
  • DDomain: (โˆ’5,8), range: {โˆ’3,4,6}
  • EDomain: [โˆ’3,6], range: {โˆ’5,4,8}

Q7:

State the domain and range of the function ๐‘“(๐‘ฅ)=๏ฒ8,โˆ’11โ‰ค๐‘ฅ<โˆ’8,4,โˆ’8<๐‘ฅ<โˆ’2,โˆ’2,โˆ’2โ‰ค๐‘ฅ<7.

  • ADomain: [โˆ’11,7]โˆ’{โˆ’8}, range: {โˆ’11,โˆ’8,7}
  • BDomain: (โˆ’11,7)โˆ’{โˆ’2}, range: {8,4,โˆ’2}
  • CDomain: [โˆ’11,7)โˆ’{โˆ’8}, range: {8,4,โˆ’2}
  • DDomain: (โˆ’11,7]โˆ’{โˆ’8}, range: {8,โˆ’8,โˆ’2}
  • EDomain: [โˆ’11,7]โˆ’{โˆ’2}, range: {8,4,โˆ’2}

Q8:

Consider the following function:

๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ7,๐‘Žโ‰ค๐‘ฅ<โˆ’1,๐‘,โˆ’1โ‰ค๐‘ฅ<3,โˆ’4,3โ‰ค๐‘ฅ<7.

If its domain is โˆ’5โ‰ค๐‘ฅ<7 and its range is {โˆ’4,2,7}, find ๐‘Ž and ๐‘.

  • A๐‘Ž=โˆ’5, ๐‘=2
  • B๐‘Ž=โˆ’4, ๐‘=2
  • C๐‘Ž=โˆ’5, ๐‘=3
  • D๐‘Ž=2, ๐‘=3
  • E๐‘Ž=2, ๐‘=โˆ’5

Q9:

Find the rule for the step function whose graph is shown.

  • A๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ9,โˆ’10<๐‘ฅโ‰คโˆ’5,5,โˆ’5<๐‘ฅ<โˆ’3,โˆ’5,โˆ’3<๐‘ฅ<2,3,2<๐‘ฅโ‰ค4
  • B๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ9,โˆ’10<๐‘ฅโ‰คโˆ’5,5,โˆ’5<๐‘ฅ<โˆ’3,โˆ’5,โˆ’3โ‰ค๐‘ฅ<2,3,2<๐‘ฅ<4
  • C๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ9,โˆ’10<๐‘ฅโ‰คโˆ’5,5,โˆ’5<๐‘ฅ<โˆ’3,โˆ’5,โˆ’3โ‰ค๐‘ฅ<2,3,2<๐‘ฅโ‰ค4
  • D๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ9,โˆ’10<๐‘ฅ<โˆ’5,5,โˆ’5<๐‘ฅ<โˆ’3,โˆ’5,โˆ’3โ‰ค๐‘ฅ<2,3,2<๐‘ฅโ‰ค4
  • E๐‘“(๐‘ฅ)=โŽงโŽชโŽจโŽชโŽฉ9,โˆ’10<๐‘ฅโ‰คโˆ’5,5,โˆ’5โ‰ค๐‘ฅ<โˆ’3,โˆ’5,โˆ’3โ‰ค๐‘ฅ<2,3,2<๐‘ฅโ‰ค4

Q10:

The function ๏ฌ‚oor(๐‘ฅ) takes a real number and rounds it down to the greatest integer less than or equal to ๐‘ฅ. For example, ๏ฌ‚oor(0.5)=0, ๏ฌ‚oor(0)=0, and ๏ฌ‚oor(โˆ’0.5)=โˆ’1.

Noah says we can just say that the floor of ๐‘ฅ is the integer part of the decimal expansion of ๐‘ฅ. Is he right? If not, why?

  • ANo. This is only true if ๐‘ฅโ‰ฅ0. For example, the floor of 0.5 is 1, not 0.
  • BNo. This is only true if ๐‘ฅโ‰ฅ0. For example, the floor of 0.5 is โˆ’1, not 0.
  • CNo. This is only true if ๐‘ฅโ‰ค0. For example, the floor of โˆ’0.5 is 0, not โˆ’1.
  • DNo. This is only true if ๐‘ฅโ‰ฅ0. For example, the floor of โˆ’0.5 is โˆ’1, not 0.
  • EYes

Define a function by ๐‘€(๐‘ฅ)=๐‘ฅโˆ’(๐‘ฅ)๏ฌ‚oor. What are the values of ๐‘€(0), ๐‘€(0.5), ๐‘€(1), and ๐‘€(โˆ’0.4)?

  • A0,0.5,1,โˆ’1.4
  • B0,0,0,0.6
  • C0,0,1,0.6
  • D0,0,1,โˆ’1.4
  • E0,0.5,0,0.6

Evidently, ๐‘€(๐‘ฅ)โ‰ฅ0. What is the range of ๐‘€?

  • Aโˆ’1โ‰ค๐‘€(๐‘ฅ)โ‰ค1
  • B0<๐‘€(๐‘ฅ)<1
  • C0<๐‘€(๐‘ฅ)โ‰ค1
  • D0โ‰ค๐‘€(๐‘ฅ)<1
  • E0โ‰ค๐‘€(๐‘ฅ)โ‰ค1

What are all the solutions to ๐‘€(๐‘ฅ)=0?

  • AAll the integers {โ€ฆ,โˆ’2,โˆ’1,0,1,2,โ€ฆ}
  • BAll the natural numbers {1,2,โ€ฆ}
  • CAll the integers {โ€ฆ,โˆ’2,โˆ’1,0,1}
  • DAll the integers {โˆ’1,0,1,2,โ€ฆ}
  • EAll the numbers {0,1,2,โ€ฆ}

The following is the graph of ๏ฌ‚oor(๐‘ฅ).

Which is the graph of ๐‘€?

  • A(d)
  • B(c)
  • C(a)
  • D(b)

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