Lesson Worksheet: Piecewise Functions Mathematics

In this worksheet, we will practice identifying, writing, and evaluating a piecewise function.

Q1:

Determine 𝑓(0).

Q2:

In the following graph, what is 𝑓(−4)?

Q3:

Determine 𝑓(0) using the graph below.

Q4:

Determine 𝑓(−3).

Q5:

Given that the function 𝑓(𝑥)=6𝑥−2𝑥<−6,−9𝑥−1−6≤𝑥≤8,−5𝑥+4𝑥>8,ififif find the value of 𝑓(4).

Q6:

Given that the function 𝑓𝑓(𝑥)=4𝑥+1,000,0≤𝑥≤4,000,8𝑥+4,600,4,000<𝑥≤30,000,8𝑥+3,400,30,000<𝑥≤90,000,: represents the price of electrical appliances in LE, where 𝑥 is the number of appliances, find 𝑓(52,000).

Q7:

Determine 𝑓(1).

  • A1
  • B5
  • Cundefined

Q8:

Determine 𝑓(2).

  • A0
  • B2
  • Cundefined
  • D−2

Q9:

Determine 𝑓(−2).

Q10:

From the following graph, determine 𝑓(2).

  • A−1
  • B0
  • C−3
  • Dundefined

Q11:

Find 𝑓(5).

  • A−2
  • B−5
  • Cundefined.
  • D5

Q12:

Given that the velocity of a motorcycle is expressed by the function 𝑣(𝑡)=5𝑡,0≤𝑡≤12,60,12<𝑡<70,−𝑡+130,70≤𝑡≤130, where 𝑡 is the time in seconds, and 𝑣 is the velocity in m/s, determine 𝑣(82).

Q13:

Consider the function 𝑀(𝑥)=𝑥,0≤𝑥<1,𝑀(𝑥+1),𝑥<0,𝑀(𝑥−1),𝑥≥1.

What are the values of 𝑀(0) and 𝑀(0.3)?

  • A𝑀(0)=−1, 𝑀(0.3)=−0.7
  • B𝑀(0)=1, 𝑀(0.3)=0.3
  • C𝑀(0)=1, 𝑀(0.3)=1.3
  • D𝑀(0)=0, 𝑀(0.3)=0.3
  • E𝑀(0)=−1, 𝑀(0.3)=−0.3

What are the values of 𝑀(1.3) and 𝑀(2.3)?

  • A𝑀(1.3)=−0.7, 𝑀(2.3)=−1.7
  • B𝑀(1.3)=0.3, 𝑀(2.3) is undefined
  • C𝑀(1.3)=1.3, 𝑀(2.3)=2.3
  • Dboth 𝑀(1.3) and 𝑀(2.3) are undefined
  • E𝑀(1.3)=0.3, 𝑀(2.3)=0.3

What are the values of 𝑀(−3) and 𝑀(678)?

  • A𝑀(−3)=0, 𝑀(678) is undefined
  • B𝑀(−3) is undefined, 𝑀(678)=0
  • Cboth 𝑀(−3) and 𝑀(678) are undefined
  • D𝑀(−3)=0, 𝑀(678)=0
  • E𝑀(−3)=−3, 𝑀(678)=678

What is the domain of the function 𝑀?

  • A[−1,2)
  • B[0,1)
  • Cℝ
  • Dℝ⧵ℤ
  • E[0,1]

Q14:

Find the missing table values for 𝑔(𝑥)=2𝑥<−2,3−2≤𝑥<3,2𝑥≥3.ififif

𝑥−303
𝑔(𝑥)
  • A18,1,8
  • B8,0,8
  • C18,0,18
  • D8,1,18

Q15:

Consider the function 𝑓(𝑥)=𝑥+4𝑥>4,2𝑥−1≤𝑥≤4,−3𝑥<−1.ififif Find 𝑓[𝑓(2)].

Q16:

Complete the table with the missing values of 𝑓(𝑥) given 𝑓(𝑥)=−3,𝑥<−7,|𝑥+4|,−7≤𝑥≤−2,2𝑥,𝑥>−2.

𝑥−8−7−5−2−1
𝑓(𝑥)
  • A−3, −3, −1, 2, −2
  • B−3, 3, 1, 2, −2
  • C3, 3, 1, −2, −2
  • D3, −3, −1, −2, 2

Q17:

The function 𝑓(𝑥)=20𝑥+4,000,1≤𝑥≤4,30𝑥+5,000,4<𝑥≤10,60𝑥+7,000,10<𝑥<15 models the sales, in dollars, of a product in a store after 𝑥 years of business.

Determine the sales after 2 years of business.

If the sales in a specific year were $4,060, what were the sales 2 years before?

Q18:

The salary of an employee is $1,500 per month. He works between 140 and 150 hours inclusive, and he gets $16 for every extra hour he works past the 150 hours.

Let 𝑥 be the number of hours he works per month and 𝑓(𝑥) be the total salary he gains per month.

Which of the following is the piecewise representation of 𝑓(𝑥)?

  • A𝑓(𝑥)=1,500,140≤𝑥≤150,1,516𝑥,𝑥>150
  • B𝑓(𝑥)=1,500,140≤𝑥≤150,1,500+16(𝑥−150),𝑥>150
  • C𝑓(𝑥)=1,500,140≤𝑥<150,1,516𝑥,𝑥≥150
  • D𝑓(𝑥)=1,500,140≤𝑥≤150,16𝑥,𝑥>150
  • E𝑓(𝑥)=1,500,140≤𝑥≤150,1,500+16𝑥,𝑥>150

Find the total salary of two months if the employee worked an extra 3 hours in the first month and an extra 2 hours in the second month.

Q19:

Find the value of 𝑎 such that 𝑓(𝑥)=3𝑥+2𝑎,𝑥≤1,𝑥+4,𝑥>1 and 𝑓(−4)=2.

Q20:

A company specialized in manufacturing clothes uses the function 𝐶(𝑥)=12,000,0<𝑥<5,12,000+2,000(𝑥−4),𝑥≥5 to determine the cost, 𝐶(𝑥), in dollars, of the number of units, 𝑥, of used fabric.

Find the cost of using 6 units of fabric.

Find the maximum number of units needed to ensure that the cost will not exceed $34,000.

Q21:

Does the graph of the function

𝑓(𝑥)=−5𝑥+1,0<𝑥≤24,3𝑥+4,𝑥>24

include the point (20,−99)?

  • ANo
  • BYes

Q22:

Consider the function 𝑓(𝑥)=−3𝑥+40≤𝑥<6,2𝑥−16≤𝑥<12,−6𝑥+9𝑥≥12.

Find the value of 𝑓(6).

Find the value of 𝑓(12).

Find the value of 𝑓(4).

Q23:

Given the function 𝑓(𝑥)=3𝑥−1,−1≤𝑥<10,2𝑥+11,𝑥≥10, find the value of 𝑎 such that 𝑓(𝑎)=2.

Q24:

Given that 𝑓(𝑥)=𝑥+12,0≤𝑥<𝑎,𝑥+34,𝑎≤𝑥≤6, find the value of 𝑎, where 𝑎∈ℤ, 𝑓(4)=16, and 𝑓(5)=59.

Q25:

The function 𝑓(𝑥) can be represented by the graph shown.

Which of the following is the piecewise representation of 𝑓(𝑥)?

  • A𝑓(𝑥)=𝑥−1−2≤𝑥<4254<𝑥≤102𝑥+8𝑥>10
  • B𝑓(𝑥)=𝑥+1−2≤𝑥<4254<𝑥<102𝑥+8𝑥≥10
  • C𝑓(𝑥)=𝑥+1−2≤𝑥<4254<𝑥≤102𝑥+8𝑥>10
  • D𝑓(𝑥)=𝑥−1−2≤𝑥<4254≤𝑥≤102𝑥+8𝑥>10
  • E𝑓(𝑥)=𝑥−1−2≤𝑥<425𝑥4<𝑥≤102𝑥+8𝑥>10

Find the value of 𝑓(3).

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