Worksheet: Graphs of Piecewise Functions

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:

Find the domain of the real function 𝑓(𝑥)=1,𝑥<7,−24,𝑥≥7.

  • A(−∞,7)
  • Bℝ
  • Cℝ−{7}
  • D(7,∞)

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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

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

Q11:

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

Q12:

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::

Q13:

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

Q14:

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

Q15:

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

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

Q16:

What kind of function is depicted in the graph?

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

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