Worksheet: Domain and Range of a Rational Function

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
  • B65
  • C56
  • D14
  • E65

Q2:

Determine the domain and the range of the function 𝑓(𝑥)=1𝑥2.

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

Q3:

Simplify the function 𝑓(𝑥)=𝑥+7𝑥8𝑥𝑥65𝑥+64, and find its domain.

  • A𝑓(𝑥)=𝑥(𝑥+1)(𝑥8), domain ={1,1,8,8}
  • B𝑓(𝑥)=𝑥(𝑥+7𝑥8)(𝑥1)(𝑥64), domain ={1,1,8,8}
  • C𝑓(𝑥)=𝑥(𝑥1)(𝑥+8), domain ={1,8}
  • D𝑓(𝑥)=𝑥(𝑥+1)(𝑥8), domain ={1,8}
  • E𝑓(𝑥)=𝑥(𝑥1)(𝑥+8), domain ={1,1,8,8}

Q4:

Find the domain of the function 𝑓(𝑥)=𝑥+4(𝑥8).

  • A{4,8}
  • B{8}
  • C{8,8}
  • D{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)=𝑥16𝑥+4.

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

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
  • B54
  • C54
  • D32
  • E12

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 {2,0}, evaluate 𝑛(3).

Q9:

Determine the domain and the range of the function 𝑓(𝑥)=1|𝑥2|.

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

Q10:

Identify the domain of 𝑛(𝑥)=9𝑥+83𝑥+2.

  • A23
  • B23
  • C32
  • D32
  • E89,23

Q11:

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

  • A76
  • B0,76
  • C0,34,76
  • D{0}
  • E34

Q12:

For which values of 𝑥 is the function 𝑛(𝑥)=𝑥25𝑥12𝑥+32 not defined?

  • A{4,8}
  • B{8,4}
  • C{5,5}
  • D{4,8}
  • E{5,5}

Q13:

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

Q14:

The function 𝑓(𝑥)=𝑥2𝑥1 has an additive inverse if the domain is .

  • A{1}
  • B{1,2}
  • C{2}
  • D{1}
  • E{2}

Q15:

If the function 𝑛(𝑥)=𝑥+3𝑥9, what is the domain of its multiplicative inverse?

  • A{9}
  • B{9,3}
  • C{3,9}
  • D{3}

Q16:

Determine the domain and the range of the function 𝑓(𝑥)=1𝑥5.

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

Q17:

Determine the domain and the range of 𝑓(𝑥)=𝑥4𝑥2.

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

Q18:

Determine the domain of the function 𝑓(𝑥)=𝑥+3|𝑥1|+7.

  • A{1}
  • B
  • C{8}
  • D{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 ={0}
  • B𝑛(𝑥)=𝑥𝑥+5, domain ={8}
  • C𝑛(𝑥)=𝑥+5𝑥, domain ={8}
  • D𝑛(𝑥)=𝑥𝑥+5, domain ={8,0}
  • E𝑛(𝑥)=𝑥+5𝑥, domain ={8,0}

Q20:

Determine the domain of the function 𝑓(𝑥)=|𝑥1||𝑥|8.

  • A{8}
  • B{0}
  • C{8}
  • D{8,8}

Q21:

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

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

Q22:

Determine the domain and the range of the function 𝑓(𝑥)=10𝑥+640𝑥64.

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

Q23:

Determine the domain and the range of the function 𝑓(𝑥)=5𝑥15𝑥+3.

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

Q24:

Given that the domain of the function 𝑛(𝑥)=8𝑥9 is the same as the domain of the function 𝑛(𝑥)=𝑥5𝑥+𝑙, what is the value of 𝑙?

Q25:

Determine the domain and the range of the function 𝑓(𝑥)=𝑥+2𝑥8𝑥+4.

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

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