Lesson Worksheet: Step Functions Mathematics

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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: −2≤𝑥<4, range: {−8,−2,6.5,4}
  • BDomain: −8<𝑥≤6.5, range: {−8,−2,6.5}
  • CDomain: −2≤𝑥<4, range: {−2,0.5,4}
  • DDomain: −8≤𝑥≤6.5, range: {−2,4}
  • EDomain: −8<𝑥≤6.5, range: {−2,0.5,4}

Q2:

State the domain and range of the following function: 𝑓(𝑥)=⎧⎪⎨⎪⎩−2,−12<𝑥≤−1112,−1<𝑥<10,1≤𝑥≤4.

  • ADomain: −12<𝑥≤4, range: −12,−1,1,112,4
  • BDomain: −2<𝑥≤112, range: {−12,−1,1,4}
  • CDomain: −2≤𝑥≤112, range: {−12,−1,1,4}
  • DDomain: −12<𝑥≤4, range: −2,112,0
  • EDomain: −12≤𝑥≤4, range: −2,112,0

Q3:

Evaluate the following function, 𝑓, for the given values of 𝑥.

𝑓(−3)

𝑓(2)

𝑓(0)

Q4:

Find the rule of the following function.

  • A𝑓(𝑥)=−1,−5≤𝑥<−23,−2≤𝑥<15,1≤𝑥≤5.
  • B𝑓(𝑥)=−1,−5≤𝑥<−23,−2≤𝑥≤15,1≤𝑥≤5.
  • C𝑓(𝑥)=−1,−5≤𝑥≤−23,−2<𝑥≤15,1≤𝑥≤5.
  • D𝑓(𝑥)=−1,−5≤𝑥≤−23,−2≤𝑥≤15,1≤𝑥≤5.
  • E𝑓(𝑥)=−1,−5<𝑥<−23,−2<𝑥<15,1<𝑥<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 𝑓(𝑥)=−3,−5<𝑥≤1,4,1<𝑥<4,6,4≤𝑥<8.

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

Q7:

State the domain and range of the function 𝑓(𝑥)=8,−11≤𝑥<−8,4,−8<𝑥<−2,−2,−2≤𝑥<7.

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

Q8:

Consider the following function:

𝑓(𝑥)=⎧⎪⎨⎪⎩7,𝑎≤𝑥<−1,𝑏,−1≤𝑥<3,−4,3≤𝑥<7.

If its domain is −5≤𝑥<7 and its range is {−4,2,7}, find 𝑎 and 𝑏.

  • A𝑎=−5, 𝑏=2
  • B𝑎=−4, 𝑏=2
  • C𝑎=−5, 𝑏=3
  • D𝑎=2, 𝑏=3
  • E𝑎=2, 𝑏=−5

Q9:

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

  • A𝑓(𝑥)=⎧⎪⎨⎪⎩9,−10<𝑥≤−5,5,−5<𝑥<−3,−5,−3<𝑥<2,3,2<𝑥≤4
  • B𝑓(𝑥)=⎧⎪⎨⎪⎩9,−10<𝑥≤−5,5,−5<𝑥<−3,−5,−3≤𝑥<2,3,2<𝑥<4
  • C𝑓(𝑥)=⎧⎪⎨⎪⎩9,−10<𝑥≤−5,5,−5<𝑥<−3,−5,−3≤𝑥<2,3,2<𝑥≤4
  • D𝑓(𝑥)=⎧⎪⎨⎪⎩9,−10<𝑥<−5,5,−5<𝑥<−3,−5,−3≤𝑥<2,3,2<𝑥≤4
  • E𝑓(𝑥)=⎧⎪⎨⎪⎩9,−10<𝑥≤−5,5,−5≤𝑥<−3,−5,−3≤𝑥<2,3,2<𝑥≤4

Q10:

The function floor(𝑥) takes a real number and rounds it down to the greatest integer less than or equal to 𝑥. For example, floor(0.5)=0, floor(0)=0, and floor(−0.5)=−1.

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

  • ANo. This is only true if 𝑥≥0. For example, the floor of 0.5 is 1, not 0.
  • BNo. This is only true if 𝑥≥0. For example, the floor of 0.5 is −1, not 0.
  • CNo. This is only true if 𝑥≤0. For example, the floor of −0.5 is 0, not −1.
  • DNo. This is only true if 𝑥≥0. For example, the floor of −0.5 is −1, not 0.
  • EYes

Define a function by 𝑀(𝑥)=𝑥−(𝑥)floor. What are the values of 𝑀(0), 𝑀(0.5), 𝑀(1), and 𝑀(−0.4)?

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

Evidently, 𝑀(𝑥)≥0. What is the range of 𝑀?

  • A−1≤𝑀(𝑥)≤1
  • B0<𝑀(𝑥)<1
  • C0<𝑀(𝑥)≤1
  • D0≤𝑀(𝑥)<1
  • E0≤𝑀(𝑥)≤1

What are all the solutions to 𝑀(𝑥)=0?

  • AAll the integers {…,−2,−1,0,1,2,…}
  • BAll the natural numbers {1,2,…}
  • CAll the integers {…,−2,−1,0,1}
  • DAll the integers {−1,0,1,2,…}
  • EAll the numbers {0,1,2,…}

The following is the graph of floor(𝑥).

Which is the graph of 𝑀?

  • A(d)
  • B(c)
  • C(a)
  • D(b)

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