Lesson Worksheet: Graphs of Piecewise Functions Mathematics

In this worksheet, we will practice graphing and analyzing a piecewise-defined function and studying its different characteristics.

Q1:

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

  • Aℝ−{−4}
  • Bℝ
  • C(−2,∞)
  • D{−2,1}

Q2:

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

  • Aℝ−[−7,7]
  • Bℝ−{−7,7}
  • Cℝ−{0}
  • Dℝ

Q3:

Find the range of the function.

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

Q4:

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

  • A(−3,∞)
  • Bℝ−{−3,2}
  • Cℝ−[−3,2]
  • D(−1,∞)
  • Eℝ−{−1}

Q5:

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

  • A𝑓(𝑥)=−1−3<𝑥<2,32≤𝑥≤3,13<𝑥≤4ififif
  • B𝑓(𝑥)=−1−3≤𝑥≤3,31<𝑥<3,13<𝑥≤4ififif
  • C𝑓(𝑥)=−1−3<𝑥<2,31≤𝑥≤3,13<𝑥≤4ififif
  • D𝑓(𝑥)=−1−3≤𝑥≤2,32<𝑥<3,13<𝑥≤4ififif
  • E𝑓(𝑥)=−1−3<𝑥<3,32≤𝑥≤3,13<𝑥≤4ififif

Q6:

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

  • Aℎ(𝑥)=𝑥2𝑥≤2,3+𝑥2≤𝑥ifif
  • Bℎ(𝑥)=𝑥2𝑥≤2,3−𝑥2≤𝑥ifif
  • Cℎ(𝑥)=3+𝑥𝑥≤2,𝑥22≤𝑥ifif
  • Dℎ(𝑥)=3−𝑥𝑥<2,𝑥22≤𝑥ifif
  • Eℎ(𝑥)=3−2𝑥𝑥≤2,𝑥22≤𝑥ifif

Q7:

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

  • A𝑝(𝑥)=𝑥2+2𝑥<2,3+𝑥2<𝑥ifif
  • B𝑝(𝑥)=3−2𝑥𝑥<2,𝑥2+22<𝑥ifif
  • C𝑝(𝑥)=3−𝑥𝑥<2,𝑥2+22<𝑥ifif
  • D𝑝(𝑥)=𝑥2+2𝑥<2,3−𝑥2<𝑥ifif
  • E𝑝(𝑥)=3+𝑥𝑥<2,𝑥2+22<𝑥ifif

Q8:

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

  • A𝑓(𝑥)=⎧⎪⎨⎪⎩𝑥2𝑥<2,2𝑥=2,3+𝑥2<𝑥ififif
  • B𝑓(𝑥)=⎧⎪⎨⎪⎩3−𝑥𝑥<2,2𝑥=2,𝑥22<𝑥ififif
  • C𝑓(𝑥)=⎧⎪⎨⎪⎩𝑥2𝑥<2,2𝑥=2,3−𝑥2<𝑥ififif
  • D𝑓(𝑥)=⎧⎪⎨⎪⎩3−2𝑥𝑥<2,2𝑥=2,𝑥22<𝑥ififif
  • E𝑓(𝑥)=⎧⎪⎨⎪⎩3+𝑥𝑥<2,2𝑥=2,𝑥22<𝑥ififif

Q9:

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

  • A𝑔(𝑥)=1𝑥<−1,(𝑥−2)(𝑥−3)1≤𝑥≤4,𝑥−24<𝑥ififif
  • B𝑔(𝑥)=1𝑥≤1,(𝑥+2)(𝑥−3)1<𝑥<4,𝑥−24≤𝑥ififif
  • C𝑔(𝑥)=1𝑥<−1,(𝑥+2)(𝑥−3)1≤𝑥≤4,𝑥−24<𝑥ififif
  • D𝑔(𝑥)=1𝑥≤1,(𝑥−2)(𝑥−3)1<𝑥<4,𝑥−24≤𝑥ififif
  • E𝑔(𝑥)=1𝑥<1,(𝑥+2)(𝑥−3)1≤𝑥≤4,𝑥−24<𝑥ififif

Q10:

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

  • A𝑥≤1𝑓(𝑥)=𝑥+4𝑥+2𝑥>1𝑓(𝑥)=𝑥+8𝑥::
  • B𝑥≤1𝑓(𝑥)=−𝑥+8𝑥𝑥>1𝑓(𝑥)=𝑥+4𝑥+2::
  • C𝑥≤1𝑓(𝑥)=𝑥+4𝑥+2𝑥>1𝑓(𝑥)=𝑥−8𝑥::
  • D𝑥≤1𝑓(𝑥)=95𝑥+365𝑥+2𝑥>1𝑓(𝑥)=−𝑥+8𝑥::
  • E𝑥≤1𝑓(𝑥)=𝑥+4𝑥+2𝑥>1𝑓(𝑥)=−𝑥+8𝑥::

Q11:

Write an equation for each part of the domains −8≤𝑥≤−2, −2<𝑥<2, 2≤𝑥<8, and 8≤𝑥≤10 of the piecewise-defined function shown in the graph.

  • A−8≤𝑥≤−2∶𝑓(𝑥)=−4−2<𝑥<2∶𝑓(𝑥)=−2𝑥−82≤𝑥<8∶𝑓(𝑥)=𝑥−6𝑥−48≤𝑥≤10∶𝑓(𝑥)=12
  • B−8≤𝑥≤−2∶𝑓(𝑥)=−4−2<𝑥<2∶𝑓(𝑥)=2𝑥−82≤𝑥<8∶𝑓(𝑥)=−𝑥+6𝑥+208≤𝑥≤10∶𝑓(𝑥)=12
  • C−8≤𝑥≤−2∶𝑓(𝑥)=−4−2<𝑥<2∶𝑓(𝑥)=−2𝑥−82≤𝑥<8∶𝑓(𝑥)=−𝑥+6𝑥+208≤𝑥≤10∶𝑓(𝑥)=12
  • D−8≤𝑥≤−2∶𝑓(𝑥)=−4−2<𝑥<2∶𝑓(𝑥)=2𝑥−82≤𝑥<8∶𝑓(𝑥)=𝑥−6𝑥−48≤𝑥≤10∶𝑓(𝑥)=12

Q12:

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

  • A𝑓(𝑥)=3𝑥−7,𝑥≤−1𝑥−1,−1<𝑥≤17𝑥−13,1<𝑥
  • B𝑓(𝑥)=3𝑥−7,𝑥≤−1𝑥+1,−1<𝑥≤27𝑥−13,2<𝑥
  • C𝑓(𝑥)=3𝑥−7,𝑥≤−1𝑥−1,−1<𝑥≤27𝑥−13,2<𝑥
  • D𝑓(𝑥)=7𝑥−3,𝑥≤−1𝑥−1,−1<𝑥≤27𝑥−13,2<𝑥
  • E𝑓(𝑥)=7𝑥−3,𝑥≤−1𝑥+1,−1<𝑥≤7𝑥−13,1<𝑥

Q13:

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

  • Aℝ−{2}
  • Bℝ
  • C(−3,∞)
  • D{−3,2}

Q14:

What kind of function is depicted in the graph?

  • Aan even function
  • Ba logarithmic function
  • Ca piecewise function
  • Da polynomial function

Q15:

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

  • A{−7}
  • Bℝ
  • C{−7,7}
  • D{7}
  • Eℝ−{−7}

Q16:

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

  • A[1,∞)
  • Bℝ−(−3,1)
  • C(−3,∞)
  • Dℝ−[−3,1)
  • E[−3,1]

Q17:

Find the range of the function.

  • Aℝ−{3,9}
  • B{3,9}
  • Cℝ−[3,9)
  • D[3,9)
  • Eℝ

Q18:

Which of the following graphs represents the function 𝑓(𝑥)=𝑥,𝑥<2−2𝑥+10,𝑥≥2?

  • A
  • B
  • C
  • D
  • E

Q19:

Consider the function graph shown.

Find the range of the function.

  • A{−3,2}
  • Bℝ−[−3,2]
  • Cℝ−[−2,3]
  • D{3,2}
  • E{−3,−2}

Find the domain of the function.

  • Aℝ−[−2,2]
  • Bℝ−[−3,2]
  • C[−3,2]
  • Dℝ−[−2,3]
  • E{−3,2}

Q20:

Which of the following is the piecewise definition of the function 𝑓(𝑥) whose graph is shown?

  • A𝑓(𝑥)=𝑥+3,−8<𝑥≤−3,𝑥−9,−3<𝑥<2,−5,2≤𝑥<6
  • B𝑓(𝑥)=𝑥+3,−8<𝑥≤−3,𝑥−9,−3<𝑥<2,−5𝑥,2≤𝑥<6
  • C𝑓(𝑥)=𝑥+3,−8<𝑥≤−3,𝑥−3,−3<𝑥<2,−5𝑥,2<𝑥<6
  • D𝑓(𝑥)=𝑥+3,−8<𝑥≤−3,𝑥−3,−3<𝑥<2,−5,2≤𝑥<6
  • E𝑓(𝑥)=𝑥+2,−8<𝑥≤−3,𝑥−9,−3<𝑥<2,−5,2≤𝑥<6

Q21:

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

  • A(−3,4)−{1}
  • B[−3,4]
  • C(−8,6)
  • Dℝ
  • E(−3,4)

Q22:

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

  • A(−∞,5)
  • Bℝ
  • Cℝ−(−∞,5]
  • D(−∞,5]
  • E[5,∞]

Q23:

Which of the following is the piecewise definition of the function 𝑓(𝑥) whose graph is shown?

  • A𝑓(𝑥)=11𝑥,−7<𝑥≤−3,𝑥+2,−3<𝑥<1
  • B𝑓(𝑥)=11,−7<𝑥≤−3,𝑥+1,−3<𝑥<1
  • C𝑓(𝑥)=11,−7<𝑥≤−3,𝑥+1,−3<𝑥≤1
  • D𝑓(𝑥)=11,−7<𝑥≤−3,𝑥+2,−3<𝑥<1
  • E𝑓(𝑥)=11,−7<𝑥<−3,𝑥+2,−3<𝑥<1

Q24:

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

  • Aℝ−[−2,3)
  • B[−2,3)
  • C[3,∞)
  • Dℝ
  • E(−∞,−2)

Q25:

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

  • A(−3,6]
  • B[−3,6)
  • Cℝ−(−3,6)
  • D[−3,6]
  • Eℝ

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