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.

  • A 5 6
  • B 6 5
  • C 5 6
  • D14
  • E 6 5

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.

  • A 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 8 ) , domain ={1,1,8,8}
  • B 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 7 𝑥 8 ) ( 𝑥 1 ) ( 𝑥 6 4 ) , domain ={1,1,8,8}
  • C 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 1 ) ( 𝑥 + 8 ) , domain ={1,8}
  • D 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) ( 𝑥 8 ) , domain ={1,8}
  • E 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 1 ) ( 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.

  • A 1 2
  • B 5 4
  • C 5 4
  • D 3 2
  • E 1 2

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.

  • A 2 3
  • B 2 3
  • C 3 2
  • D 3 2
  • E 8 9 , 2 3

Q11:

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

  • A 7 6
  • B 0 , 7 6
  • C 0 , 3 4 , 7 6
  • D { 0 }
  • E 3 4

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.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , domain ={0}
  • B 𝑛 ( 𝑥 ) = 𝑥 𝑥 + 5 , domain ={8}
  • C 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , domain ={8}
  • D 𝑛 ( 𝑥 ) = 𝑥 𝑥 + 5 , domain ={8,0}
  • E 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 , 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}.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.