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Lesson: Graphing Piecewise Functions

In this lesson, we will learn how to analyze a piecewise-defined function, graph it, find its domain and range, and write the equation that represents it from its graph.

Sample Question Videos

01:25

Worksheet • 17 Questions • 1 Video

Q1:

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

  • A ℝ − { − 4 }
  • B ℝ
  • C ] − 2 , ∞ [
  • D { − 2 , 1 }

Q2:

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

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

Q3:

Find the range of the function.

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

Q4:

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

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

Q5:

Find the domain of the real function

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

Q6:

Find the domain of the real function

  • A ℝ
  • B ℝ − { 1 }
  • C ] 1 , ∞ [
  • D ] − ∞ , 1 [

Q7:

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

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

Q9:

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

  • A 𝑝 ( 𝑥 ) = 3 − 𝑥 𝑥 < 2 , 𝑥 2 + 2 2 < 𝑥 i f i f
  • B 𝑝 ( 𝑥 ) = 3 − 2 𝑥 𝑥 < 2 , 𝑥 2 + 2 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 ) = 𝑥 2 + 2 𝑥 < 2 , 3 + 𝑥 2 < 𝑥 i f i f
  • E 𝑝 ( 𝑥 ) = 𝑥 2 + 2 𝑥 < 2 , 3 − 𝑥 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 − 2 𝑥 𝑥 < 2 , 2 𝑥 = 2 , 𝑥 2 2 < 𝑥 i f i f i f
  • C 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 3 + 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 , 𝑑 a n d are numbers that you should find.

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

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

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

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 − 4 8 ≤ 𝑥 ≤ 1 0 𝑓 ( 𝑥 ) = 1 2 : : : : 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 : : : : 2
  • C − 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 : : : : 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 : : : : 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 ) = 7 𝑥 − 3 , 𝑥 ≤ − 1 𝑥 − 1 , − 1 < 𝑥 ≤ 2 7 𝑥 − 1 3 , 2 < 𝑥
  • C 𝑓 ( 𝑥 ) = 3 𝑥 − 7 , 𝑥 ≤ − 1 𝑥 − 1 , − 1 < 𝑥 ≤ 1 7 𝑥 − 1 3 , 1 < 𝑥
  • D 𝑓 ( 𝑥 ) = 7 𝑥 − 3 , 𝑥 ≤ − 1 𝑥 + 1 , − 1 < 𝑥 ≤ 7 𝑥 − 1 3 , 1 < 𝑥
  • E 𝑓 ( 𝑥 ) = 3 𝑥 − 7 , 𝑥 ≤ − 1 𝑥 + 1 , − 1 < 𝑥 ≤ 2 7 𝑥 − 1 3 , 2 < 𝑥

Q17:

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

  • A ℝ − { 2 }
  • B ℝ
  • C ] − 3 , ∞ [
  • D { − 3 , 2 }
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