Worksheet: Graphing Piecewise Functions

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In this worksheet, we will practice analyzing a piecewise-defined function, graph it, find its domain and range, and write the equation that represents it from its graph.

Q1:

Determine the domain of the function represented by the given graph.

  • A ( − 2 , ∞ )
  • B { − 2 , 1 }
  • C ℝ − { − 4 }
  • D ℝ

Q2:

Determine the domain of the function represented by the given graph.

  • A ℝ
  • B ℝ − { − 7 , 7 }
  • C ℝ − [ − 7 , 7 ]
  • D ℝ − { 0 }

Q3:

Find the range of the function.

  • A ( − ∞ , − 2 ]
  • B ℝ
  • C [ − 1 , ∞ )
  • D [ − 2 , ∞ )
  • E ( − ∞ , − 1 ]

Q4:

Determine the range of the function represented by the following graph.

  • A ℝ − [ − 3 , 2 ]
  • B ℝ − { − 3 , 2 }
  • C ( − 1 , ∞ )
  • D ℝ − { − 1 }
  • E ( − 3 , ∞ )

Q5:

Find the domain of the real function 𝑓 ( 𝑥 ) = 1 𝑥 < 7 , − 2 4 𝑥 ≥ 7 . i f i f

  • A ℝ
  • B ( 7 , ∞ )
  • C ℝ − { 7 }
  • D ( − ∞ , 7 )

Q6:

Find the domain of the real function 𝑓 ( 𝑥 ) = − 5 𝑥 ≤ 1 , 2 5 − 𝑥 𝑥 > 1 . i f i f

  • A ℝ − { 1 }
  • B ( 1 , ∞ )
  • C ( − ∞ , 1 )
  • D ℝ

Q7:

Give the piecewise definition of the function 𝑓 whose graph is shown.

  • A 𝑓 ( 𝑥 ) = − 1 − 3 < 𝑥 < 3 , 3 2 ≤ 𝑥 ≤ 3 , 1 3 < 𝑥 ≤ 4 i f i f i f
  • B 𝑓 ( 𝑥 ) = − 1 − 3 < 𝑥 < 2 , 3 1 ≤ 𝑥 ≤ 3 , 1 3 < 𝑥 ≤ 4 i f i f i f
  • C 𝑓 ( 𝑥 ) = − 1 − 3 ≤ 𝑥 ≤ 2 , 3 2 < 𝑥 < 3 , 1 3 < 𝑥 ≤ 4 i f i f i f
  • D 𝑓 ( 𝑥 ) = − 1 − 3 < 𝑥 < 2 , 3 2 ≤ 𝑥 ≤ 3 , 1 3 < 𝑥 ≤ 4 i f i f i f
  • E 𝑓 ( 𝑥 ) = − 1 − 3 ≤ 𝑥 ≤ 3 , 3 1 < 𝑥 < 3 , 1 3 < 𝑥 ≤ 4 i f i f i f

Q8:

Give the piecewise definition of the function ℎ whose graph is shown.

  • A ℎ ( 𝑥 ) = 3 + 𝑥 𝑥 ≤ 2 , 𝑥 2 2 ≤ 𝑥 i f i f
  • B ℎ ( 𝑥 ) = 𝑥 2 𝑥 ≤ 2 , 3 − 𝑥 2 ≤ 𝑥 i f i f
  • C ℎ ( 𝑥 ) = 3 − 2 𝑥 𝑥 ≤ 2 , 𝑥 2 2 ≤ 𝑥 i f i f
  • D ℎ ( 𝑥 ) = 𝑥 2 𝑥 ≤ 2 , 3 + 𝑥 2 ≤ 𝑥 i f i f
  • E ℎ ( 𝑥 ) = 3 − 𝑥 𝑥 < 2 , 𝑥 2 2 ≤ 𝑥 i f i f

Q9:

Give the piecewise definition of the function 𝑝 whose graph is shown.

  • A 𝑝 ( 𝑥 ) = 𝑥 2 + 2 𝑥 < 2 , 3 − 𝑥 2 < 𝑥 i f i f
  • B 𝑝 ( 𝑥 ) = 𝑥 2 + 2 𝑥 < 2 , 3 + 𝑥 2 < 𝑥 i f i f
  • C 𝑝 ( 𝑥 ) = 3 − 𝑥 𝑥 < 2 , 𝑥 2 + 2 2 < 𝑥 i f i f
  • D 𝑝 ( 𝑥 ) = 3 + 𝑥 𝑥 < 2 , 𝑥 2 + 2 2 < 𝑥 i f i f
  • E 𝑝 ( 𝑥 ) = 3 − 2 𝑥 𝑥 < 2 , 𝑥 2 + 2 2 < 𝑥 i f i f

Q10:

Give the piecewise definition of the function 𝑓 whose graph is shown.

  • A 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 3 + 𝑥 𝑥 < 2 , 2 𝑥 = 2 , 𝑥 2 2 < 𝑥 i f i f i f
  • B 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 3 − 𝑥 𝑥 < 2 , 2 𝑥 = 2 , 𝑥 2 2 < 𝑥 i f i f i f
  • C 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 3 − 2 𝑥 𝑥 < 2 , 2 𝑥 = 2 , 𝑥 2 2 < 𝑥 i f i f i f
  • D 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 2 𝑥 < 2 , 2 𝑥 = 2 , 3 + 𝑥 2 < 𝑥 i f i f i f
  • E 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 2 𝑥 < 2 , 2 𝑥 = 2 , 3 − 𝑥 2 < 𝑥 i f i f i f

Q11:

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 + 𝑏 𝑥 + 𝑐 | 𝑥 + 1 | + 𝑑 | 𝑥 − 2 | , where 𝑎 , 𝑏 , 𝑐 , and 𝑑 are numbers that you should find.

  • A 𝑓 ( 𝑥 ) = − 9 + 𝑥 + | 𝑥 + 1 | + 3 | 𝑥 − 2 |
  • B 𝑓 ( 𝑥 ) = 9 + 𝑥 + 2 | 𝑥 + 1 | + 3 | 𝑥 − 2 |
  • C 𝑓 ( 𝑥 ) = − 9 − 2 𝑥 − 2 | 𝑥 + 1 | + 3 | 𝑥 − 2 |
  • D 𝑓 ( 𝑥 ) = 9 + 2 𝑥 + | 𝑥 + 1 | + 3 | 𝑥 − 2 |
  • E 𝑓 ( 𝑥 ) = − 9 + 2 𝑥 + 2 | 𝑥 + 1 | + 3 | 𝑥 − 2 |

Q12:

Give the piecewise definition of the function 𝑔 whose graph is shown.

  • A 𝑔 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝑥 < 1 , ( 𝑥 + 2 ) ( 𝑥 − 3 ) 1 ≤ 𝑥 ≤ 4 , 𝑥 − 2 4 < 𝑥 i f i f i f
  • B 𝑔 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝑥 ≤ 1 , ( 𝑥 + 2 ) ( 𝑥 − 3 ) 1 < 𝑥 < 4 , 𝑥 − 2 4 ≤ 𝑥 i f i f i f
  • C 𝑔 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝑥 < − 1 , ( 𝑥 + 2 ) ( 𝑥 − 3 ) 1 ≤ 𝑥 ≤ 4 , 𝑥 − 2 4 < 𝑥 i f i f i f
  • D 𝑔 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝑥 < − 1 , ( 𝑥 − 2 ) ( 𝑥 − 3 ) 1 ≤ 𝑥 ≤ 4 , 𝑥 − 2 4 < 𝑥 i f i f i f
  • E 𝑔 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝑥 ≤ 1 , ( 𝑥 − 2 ) ( 𝑥 − 3 ) 1 < 𝑥 < 4 , 𝑥 − 2 4 ≤ 𝑥 i f i f i f

Q13:

The graph in figure (i) is of 𝑓 ( 𝑥 ) = 5 2 | 𝑥 | + 1 2 𝑥 , which could also be written as follows 𝑓 ( 𝑥 ) = 3 𝑥 , 𝑥 ≥ 0 − 2 𝑥 , 𝑥 < 0 .

Find the values of 𝑎 and 𝑏 that would make graph (ii) that of 𝑔 ( 𝑥 ) = 𝑎 | 𝑥 − 5 | + 𝑏 ( 𝑥 − 5 ) .

  • A 𝑎 = 3 , 𝑏 = − 1
  • B 𝑎 = 3 , 𝑏 = 1
  • C 𝑎 = 1 , 𝑏 = 5
  • D 𝑎 = 2 , 𝑏 = 1
  • E 𝑎 = 2 , 𝑏 = − 1

Q14:

Write an equation for each part of the domains 𝑥 ≤ 1 and 𝑥 > 1 of the piecewise-defined function shown in the graph.

  • A 𝑥 ≤ 1 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 2 𝑥 > 1 𝑓 ( 𝑥 ) = 𝑥 − 8 𝑥 : :
  • B 𝑥 ≤ 1 𝑓 ( 𝑥 ) = 9 5 𝑥 + 3 6 5 𝑥 + 2 𝑥 > 1 𝑓 ( 𝑥 ) = − 𝑥 + 8 𝑥 : :
  • C 𝑥 ≤ 1 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 2 𝑥 > 1 𝑓 ( 𝑥 ) = − 𝑥 + 8 𝑥 : :
  • D 𝑥 ≤ 1 𝑓 ( 𝑥 ) = − 𝑥 + 8 𝑥 𝑥 > 1 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 2 : :
  • E 𝑥 ≤ 1 𝑓 ( 𝑥 ) = 𝑥 + 4 𝑥 + 2 𝑥 > 1 𝑓 ( 𝑥 ) = 𝑥 + 8 𝑥 : :

Q15:

Write an equation for each part of the domains − 8 ≤ 𝑥 ≤ − 2 , − 2 < 𝑥 < 2 , 2 ≤ 𝑥 < 8 , and 8 ≤ 𝑥 ≤ 1 0 of the piecewise-defined function shown in the graph.

  • A − 8 ≤ 𝑥 ≤ − 2 ∶ 𝑓 ( 𝑥 ) = − 4 − 2 < 𝑥 < 2 ∶ 𝑓 ( 𝑥 ) = − 2 𝑥 − 8 2 ≤ 𝑥 < 8 ∶ 𝑓 ( 𝑥 ) = − 𝑥 + 6 𝑥 + 2 0 8 ≤ 𝑥 ≤ 1 0 ∶ 𝑓 ( 𝑥 ) = 1 2
  • B − 8 ≤ 𝑥 ≤ − 2 ∶ 𝑓 ( 𝑥 ) = − 4 − 2 < 𝑥 < 2 ∶ 𝑓 ( 𝑥 ) = − 2 𝑥 − 8 2 ≤ 𝑥 < 8 ∶ 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 − 4 8 ≤ 𝑥 ≤ 1 0 ∶ 𝑓 ( 𝑥 ) = 1 2
  • C − 8 ≤ 𝑥 ≤ − 2 ∶ 𝑓 ( 𝑥 ) = − 4 − 2 < 𝑥 < 2 ∶ 𝑓 ( 𝑥 ) = 2 𝑥 − 8 2 ≤ 𝑥 < 8 ∶ 𝑓 ( 𝑥 ) = 𝑥 − 6 𝑥 − 4 8 ≤ 𝑥 ≤ 1 0 ∶ 𝑓 ( 𝑥 ) = 1 2
  • D − 8 ≤ 𝑥 ≤ − 2 ∶ 𝑓 ( 𝑥 ) = − 4 − 2 < 𝑥 < 2 ∶ 𝑓 ( 𝑥 ) = 2 𝑥 − 8 2 ≤ 𝑥 < 8 ∶ 𝑓 ( 𝑥 ) = − 𝑥 + 6 𝑥 + 2 0 8 ≤ 𝑥 ≤ 1 0 ∶ 𝑓 ( 𝑥 ) = 1 2

Q16:

Which of the following is the function whose graph is shown?

  • A 𝑓 ( 𝑥 ) = 3 𝑥 − 7 , 𝑥 ≤ − 1 𝑥 + 1 , − 1 < 𝑥 ≤ 2 7 𝑥 − 1 3 , 2 < 𝑥
  • B 𝑓 ( 𝑥 ) = 3 𝑥 − 7 , 𝑥 ≤ − 1 𝑥 − 1 , − 1 < 𝑥 ≤ 1 7 𝑥 − 1 3 , 1 < 𝑥
  • C 𝑓 ( 𝑥 ) = 3 𝑥 − 7 , 𝑥 ≤ − 1 𝑥 − 1 , − 1 < 𝑥 ≤ 2 7 𝑥 − 1 3 , 2 < 𝑥
  • D 𝑓 ( 𝑥 ) = 7 𝑥 − 3 , 𝑥 ≤ − 1 𝑥 − 1 , − 1 < 𝑥 ≤ 2 7 𝑥 − 1 3 , 2 < 𝑥
  • E 𝑓 ( 𝑥 ) = 7 𝑥 − 3 , 𝑥 ≤ − 1 𝑥 + 1 , − 1 < 𝑥 ≤ 7 𝑥 − 1 3 , 1 < 𝑥

Q17:

Determine the domain of the function represented by the given graph.

  • A { − 3 , 2 }
  • B ℝ
  • C ( − 3 , ∞ )
  • D ℝ − { 2 }

Q18:

What kind of function is depicted in the graph?

  • A an even function
  • B a logarithmic function
  • Ca piecewise function
  • D a polynomial function

Q19:

Let ℎ ( 𝑡 ) be the level of lake Malawi’s water at a fixed location as a function of the date 𝑡 in the year 2016. The date is measured from January 1, and the level is measured in centimeters above or below the mark set on January 1. Given that the lake is filled by rain that falls between November and April and that the Shire River is the sole outlet, which of the following is a possible graph of ℎ ?

  • A
  • B
  • C
  • D
  • E

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