Lesson Worksheet: Absolute Value Functions Mathematics

In this worksheet, we will practice evaluating and graphing absolute value functions and identifying their domain and range.

Q1:

Determine the domain and the range of the function 𝑓(𝑥)=|𝑥1|+1.

  • AThe domain is and the range is [1,).
  • BThe domain is [1,) and the range is .
  • CThe domain is (1,) and the range is {1}.
  • DThe domain is {1} and the range is (1,).

Q2:

Find the range of the function 𝑓(𝑥)=|2𝑥2|.

  • A{1}
  • B{1}
  • C
  • D[1,)
  • E[0,)

Q3:

Find the range of the function 𝑓(𝑥)=|2𝑥+1|.

  • A
  • B[0,)
  • C12
  • D12,

Q4:

Find the domain of the function 𝑓(𝑥)=|3𝑥8|.

  • A[0,)
  • B83,
  • C83
  • D

Q5:

Find the domain and range of the function 𝑓(𝑥)=|3𝑥1|+3𝑥+5.

  • AThe domain is , and the range is 13,.
  • BThe domain is , and the range is [6,).
  • CThe domain is 13,, and the range is [6,).
  • DThe domain is 13,, and the range is .

Q6:

Determine the domain and the range of the function 𝑓(𝑥)=|5𝑥+5|9.

  • AThe domain is (9,), and the range is {1}.
  • BThe domain is [9,), and the range is .
  • CThe domain is , and the range is [9,).
  • DThe domain is {1}, and the range is (9,).
  • EThe domain is , and the range is [0,).

Q7:

Given that 𝑎 is a constant, what is the domain of the function 𝑓(𝑥)=|𝑥+𝑎|?

  • A(,𝑎]
  • B[𝑎,)
  • C
  • D𝑎

Q8:

Find the domain and range of the function 𝑓(𝑥)=4|𝑥5|1.

  • AThe domain is (,1] and the range is .
  • BThe domain is {5} and the range is {1}.
  • CThe domain is and the range is (,1).
  • DThe domain is and the range is (,1].

Q9:

Given that 𝑎 is a constant, what is the domain of the function 𝑓(𝑥)=|𝑥𝑎|?

  • A
  • B𝑎
  • C[𝑎,)
  • D(,𝑎]

Q10:

Find the domain of the function 𝑓(𝑥)=|8𝑥+7|.

  • A[0,)
  • B78
  • C78,
  • D
  • E78

Q11:

Determine the domain and the range of the function 𝑓(𝑥)=|𝑥+1|+1.

  • AThe domain is and the range is [1,).
  • BThe domain is [1,) and the range is .
  • CThe domain is (1,) and the range is {1}.
  • DThe domain is {1} and the range is (1,).

Q12:

Find the range of the function 𝑓(𝑥)=|6𝑥4|.

  • A
  • B[0,)
  • C23
  • D23,

Q13:

Find the range of the function 𝑓(𝑥)=|𝑥+8|.

  • A
  • B[0,)
  • C{8}
  • D[8,)

Q14:

Find the range of the function 𝑓(𝑥)=|2𝑥+1|.

  • A12
  • B12
  • C
  • D12,
  • E[0,)

Q15:

Find the range of the function 𝑓(𝑥)=|2𝑥+2|.

  • A{1}
  • B{1}
  • C
  • D[1,)
  • E[0,)

Q16:

Find the domain of the function 𝑓(𝑥)=|5𝑥+5|.

  • A[0,)
  • B[1,)
  • C{1}
  • D

Q17:

Find the domain of the function 𝑓(𝑥)=|4𝑥+5|.

  • A[0,)
  • B54,
  • C54
  • D

Q18:

A body was moving with a uniform velocity of magnitude 5 cm/s from the point 𝐴 to the point 𝐶 passing through the point 𝐵 without stopping. The distance between the body and the point 𝐵 is given by 𝑑(𝑡)=5|8𝑡|, where 𝑡 is the time in seconds, and 𝑑 is the distance in cm. Determine the distance between the body and the point 𝐵 after 5 seconds and after 11 seconds.

  • A15 cm, 15 cm
  • B3 cm, 40 cm
  • C5 cm, 5 cm
  • D15 cm, 5 cm

Q19:

A body moved from position 𝐴 to position 𝐶 passing through position 𝐵 with a uniform velocity of 3 cm/s and without stopping. If the distance between the body and position 𝐵 is given by 𝑑(𝑡)=3|7𝑡|, where 𝑡 is the time in seconds, and 𝑑 is the distance in centimeters, determine the time interval during which the body is less than 9 cm from 𝐵.

  • A[4,10]
  • B(4,10)
  • C(4,)
  • D(10,)

Q20:

Find the horizontal intercepts of the function 𝑓(𝑥)=2|𝑥+2|10.

  • A𝑥=3,7
  • B𝑥=3,7
  • C𝑥=5,5
  • DNo horizontal intercepts

Q21:

The graph in figure (i) is of 𝑓(𝑥)=52|𝑥|+12𝑥, which could also be written as follows 𝑓(𝑥)=3𝑥,𝑥02𝑥,𝑥<0.

Find the values of 𝑎 and 𝑏 that would make graph (ii) that of 𝑔(𝑥)=𝑎|𝑥5|+𝑏(𝑥5).

  • A𝑎=3,𝑏=1
  • B𝑎=1,𝑏=5
  • C𝑎=2,𝑏=1
  • D𝑎=3,𝑏=1
  • E𝑎=2,𝑏=1

Q22:

The graph of the function 𝑓 is formed of a ray with slope 3 from the point (1,2), a line segment between the points (1,2) and (2,1), and a ray with slope 7 from the point (2,1). Write the function in the form 𝑓(𝑥)=𝑎+𝑏𝑥+𝑐|𝑥+1|+𝑑|𝑥2|, where 𝑎, 𝑏, 𝑐, and 𝑑 are numbers that you should find.

  • A𝑓(𝑥)=92𝑥2|𝑥+1|+3|𝑥2|
  • B𝑓(𝑥)=9+2𝑥+|𝑥+1|+3|𝑥2|
  • C𝑓(𝑥)=9+2𝑥+2|𝑥+1|+3|𝑥2|
  • D𝑓(𝑥)=9+𝑥+|𝑥+1|+3|𝑥2|
  • E𝑓(𝑥)=9+𝑥+2|𝑥+1|+3|𝑥2|

Q23:

Which of the following are the correct range and domain of 𝑓(𝑥)=|𝑥+2|+5?

  • AThe domain is , and the range is (5,).
  • BThe domain is , and the range is [5,).
  • CThe domain is [5,), and the range is .
  • DThe domain is , and the range is [2,).
  • EThe domain is (5,), and the range is .

Q24:

Which of the following is the range of 𝑓(𝑥)=|3𝑥5|+7?

  • A(7,)
  • B
  • C[7,)
  • D{7}
  • E[5,)

Q25:

Which of the following functions has the domain and the range [3,)?

  • A𝑓(𝑥)=|𝑥+2|
  • B𝑓(𝑥)=|𝑥+2|3
  • C𝑓(𝑥)=|𝑥+3|2
  • D𝑓(𝑥)=|𝑥+2|+3
  • E𝑓(𝑥)=|𝑥3|+2

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