Worksheet: Domain and Range of a Rational Function

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In this worksheet, we will practice finding the domain and range of a rational function either from its graph or its defining rule.

Q1:

Find the value of 𝑐 given 𝑛(𝑥)=1425𝑥+60𝑥+36 where 𝑛(𝑐) is undefined.

  • A56
  • B−65
  • C−56
  • D14
  • E65

Q2:

Determine the domain and the range of the function 𝑓(𝑥)=1𝑥−2.

  • AThe domain is ℝ−{2}, and the range is ℝ−{0}.
  • BThe domain is {0}, and the range is ℝ.
  • CThe domain is ℝ, and the range is ℝ.
  • DThe domain is ℝ, and the range is {2}.

Q3:

Simplify the function 𝑓(𝑥)=𝑥+7𝑥−8𝑥𝑥−65𝑥+64, and find its domain.

  • A𝑓(𝑥)=𝑥(𝑥+1)(𝑥−8), domain =ℝ−{1,−1,−8,8}
  • B𝑓(𝑥)=𝑥(𝑥+7𝑥−8)(𝑥−1)(𝑥−64), domain =ℝ−{1,−1,−8,8}
  • C𝑓(𝑥)=𝑥(𝑥−1)(𝑥+8), domain =ℝ−{1,−8}
  • D𝑓(𝑥)=𝑥(𝑥+1)(𝑥−8), domain =ℝ−{−1,8}
  • E𝑓(𝑥)=𝑥(𝑥−1)(𝑥+8), domain =ℝ−{1,−1,−8,8}

Q4:

Find the domain of the function 𝑓(𝑥)=𝑥+4(𝑥−8).

  • Aℝ−{−4,8}
  • Bℝ−{8}
  • Cℝ−{−8,8}
  • Dℝ−{−4}

Q5:

The domain of an algebraic fractional function is the set of all real numbers except the .

  • Azeros of the function
  • Bzeros of the denominator of the function

Q6:

Determine the domain and the range of the function 𝑓(𝑥)=𝑥−16𝑥+4.

  • AThe domain is {−4} and the range is ℝ.
  • BThe domain is ℝ−{−4} and the range is ℝ−{−8}.
  • CThe domain is ℝ and the range is {−4}.
  • DThe domain is ℝ−{−8} and the range is ℝ−{−4}.
  • EThe domain is {−8,−4} and the range is ℝ.

Q7:

Define a function on real numbers by 𝑓(𝑥)=2𝑥+34𝑥+5.

What is the domain of this function?

  • Aall real numbers except 54
  • Ball real numbers except 32
  • Call real numbers except −54
  • Dall real numbers except −32
  • Eall real numbers

Find the one value that 𝑓(𝑥) cannot take.

  • A12
  • B−54
  • C54
  • D−32
  • E−12

What is the range of this function?

  • Aall real numbers except −32
  • Ball real numbers
  • Call real numbers except −54
  • Dall real numbers except 12
  • Eall real numbers except −12

Q8:

Given that the domain of the function 𝑛(𝑥)=36𝑥+20𝑥+𝑎 is ℝ−{−2,0}, evaluate 𝑛(3).

Q9:

Determine the domain and the range of the function 𝑓(𝑥)=1|𝑥−2|.

  • AThe domain is (0,∞) and the range is ℝ−{2}.
  • BThe domain is {2} and the range is ℝ.
  • CThe domain is ℝ and the range is {2}.
  • DThe domain is ℝ−{2} and the range is (0,∞).

Q10:

Identify the domain of 𝑛(𝑥)=9𝑥+83𝑥+2.

  • Aℝ−−23
  • Bℝ−23
  • Cℝ−−32
  • Dℝ−32
  • Eℝ−−89,−23

Q11:

Determine the domain of the function 𝑛(𝑥)=4𝑥+3𝑥6𝑥+7𝑥.

  • Aℝ−−76
  • Bℝ−0,−76
  • Cℝ−0,−34,−76
  • Dℝ−{0}
  • Eℝ−−34

Q12:

For which values of 𝑥 is the function 𝑛(𝑥)=𝑥−25𝑥−12𝑥+32 not defined?

  • Aℝ−{4,8}
  • B{−8,−4}
  • C{−5,5}
  • D{4,8}
  • Eℝ−{−5,5}

Q13:

If the common domain of the two functions 𝑛(𝑥)=3𝑥+3 and 𝑛(𝑥)=3𝑥𝑥−𝑚 is ℝ−{−3,4}, what is the value of 𝑚?

Q14:

The function 𝑓(𝑥)=𝑥−2𝑥−1 has an additive inverse if the domain is .

  • Aℝ−{−1}
  • Bℝ−{1,2}
  • Cℝ−{−2}
  • Dℝ−{1}
  • Eℝ−{2}

Q15:

If the function 𝑛(𝑥)=𝑥+3𝑥−9, what is the domain of its multiplicative inverse?

  • Aℝ−{9}
  • Bℝ−{−9,3}
  • Cℝ−{−3,9}
  • Dℝ−{−3}

Q16:

Determine the domain and the range of the function 𝑓(𝑥)=1𝑥−5.

  • AThe domain is ℝ−{5}, the range is ℝ−{0}.
  • BThe domain is ℝ−{5}, the range is ℝ−{5}.
  • CThe domain is ℝ−{0}, the range is ℝ−{5}.
  • DThe domain is ℝ−{0}, the range is ℝ−{0}.

Q17:

Determine the domain and the range of 𝑓(𝑥)=𝑥−4𝑥−2.

  • AThe domain is {2,4} and the range is ℝ.
  • BThe domain is ℝ−{2} and the range is ℝ−{4}.
  • CThe domain is ℝ−{4} and the range is ℝ−{2}.
  • DThe domain is {2} and the range is ℝ.
  • EThe domain is ℝ and the range is {2}.

Q18:

Determine the domain of the function 𝑓(𝑥)=𝑥+3|𝑥−1|+7.

  • Aℝ−{1}
  • Bℝ
  • Cℝ−{8}
  • Dℝ−{−8,8}

Q19:

Given the function 𝑛(𝑥)=𝑥+8𝑥(𝑥+8)(𝑥+5), find the multiplicative inverse of 𝑛 in its simplest form and state its domain.

  • A𝑛(𝑥)=𝑥+5𝑥, domain =ℝ−{0}
  • B𝑛(𝑥)=𝑥𝑥+5, domain =ℝ−{−8}
  • C𝑛(𝑥)=𝑥+5𝑥, domain =ℝ−{−8}
  • D𝑛(𝑥)=𝑥𝑥+5, domain =ℝ−{−8,0}
  • E𝑛(𝑥)=𝑥+5𝑥, domain =ℝ−{−8,0}

Q20:

Determine the domain of the function 𝑓(𝑥)=|𝑥−1||𝑥|−8.

  • Aℝ−{−8}
  • Bℝ−{0}
  • Cℝ−{8}
  • Dℝ−{−8,8}

Q21:

Determine the domain and the range of the function 𝑓(𝑥)=𝑥+9𝑥+20𝑥+5.

  • AThe domain is ℝ−{−5} and the range is ℝ−{−1}.
  • BThe domain is ℝ and the range is {−5}.
  • CThe domain is ℝ−{−1} and the range is ℝ−{−5}.
  • DThe domain is {−5} and the range is ℝ.

Q22:

Determine the domain and the range of the function 𝑓(𝑥)=−10𝑥+640𝑥−64.

  • AThe domain is ℝ, and the range is {−10}.
  • BThe domain is {−10}, and the range is ℝ−{−8,8}.
  • CThe domain is ℝ−{−8,8}, and the range is {−10}.
  • DThe domain is {−10}, and the range is ℝ.
  • EThe domain is {−8,8}, and the range is ℝ.

Q23:

Determine the domain and the range of the function 𝑓(𝑥)=−5𝑥−15𝑥+3.

  • AThe domain is {−5} and the range is ℝ−{−3}.
  • BThe domain is {−5} and the range is ℝ.
  • CThe domain is ℝ−{−3} and the range is {−5}.
  • DThe domain is ℝ and the range is {−5}.

Q24:

Given that the domain of the function 𝑛(𝑥)=8𝑥−9 is the same as the domain of the function 𝑛(𝑥)=𝑥−5𝑥+𝑙, what is the value of 𝑙?

Q25:

Determine the domain and the range of the function 𝑓(𝑥)=𝑥+2𝑥−8𝑥+4.

  • AThe domain is ℝ and the range is {−4}.
  • BThe domain is ℝ−{−6} and the range is ℝ−{−4}.
  • CThe domain is {−6,−4} and the range is ℝ.
  • DThe domain is {−4} and the range is ℝ.
  • EThe domain is ℝ−{−4} and the range is ℝ−{−6}.

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