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

Start Practicing

Worksheet: Composite Functions

Q1:

Given 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 1 and 𝑔 ( π‘₯ ) = π‘₯ + 1 2 , find ( 𝑓 ∘ 𝑔 ) ( 2 ) .

Q2:

Given that the function 𝑓 ( π‘₯ ) = 1 9 π‘₯ 2 and the function 𝑔 ( π‘₯ ) = βˆ’ 2 π‘₯ , determine ( 𝑔 ∘ 𝑓 ) ( π‘₯ ) in its simplest form, and evaluate ( 𝑔 ∘ 𝑓 ) ( 1 ) .

  • A 3 8 π‘₯ 2 , ( 𝑔 ∘ 𝑓 ) ( 1 ) = 3 8
  • B 7 6 π‘₯ 2 , ( 𝑔 ∘ 𝑓 ) ( 1 ) = 7 6
  • C 1 9 π‘₯ 2 , ( 𝑔 ∘ 𝑓 ) ( 1 ) = 1 9
  • D βˆ’ 3 8 π‘₯ 2 , ( 𝑔 ∘ 𝑓 ) ( 1 ) = βˆ’ 3 8

Q3:

If 𝑓 ( π‘₯ ) = 3 βˆ’ π‘₯ 2 and 𝑔 ( π‘₯ ) = 2 π‘₯ + 4 , find ( 𝑓 ∘ 𝑔 ) ( 1 ) .

Q4:

Given that the function , and the function , determine in its simplest form, and find its domain.

  • A , domain
  • B , domain
  • C , domain
  • D , domain
  • E , domain

Q5:

Given that the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 9 2 , and the function 𝑔 ( π‘₯ ) = √ π‘₯ + 1 7 , find ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) in its simplest form, then determine ( 𝑓 ∘ 𝑔 ) ( 1 9 ) .

  • A ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = π‘₯ + 1 0 6 , ( 𝑓 ∘ 𝑔 ) ( 1 9 ) = 1 2 5
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = √ π‘₯ βˆ’ 7 2 2 , ( 𝑓 ∘ 𝑔 ) ( 1 9 ) = 1 7
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = π‘₯ βˆ’ 1 0 6 , ( 𝑓 ∘ 𝑔 ) ( 1 9 ) = βˆ’ 8 7
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = π‘₯ βˆ’ 7 2 , ( 𝑓 ∘ 𝑔 ) ( 1 9 ) = βˆ’ 5 3
  • E ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = ο€» √ π‘₯ + 1 7  βˆ’ 8 9 , ( 𝑓 ∘ 𝑔 ) ( 1 9 ) = βˆ’ 8 3

Q6:

Given that the function , and the function , express in its simplest form, and find its domain, then evaluate .

  • A , domain ,
  • B , domain ,
  • C , domain ,
  • D , domain ,
  • E , domain ,

Q7:

For 𝑓 ( π‘₯ ) = 3 π‘₯ and 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 2 , express ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) in the form 𝐴 𝑏 π‘₯ with suitable numbers for 𝐴 and 𝑏 .

  • A ( π‘₯ βˆ’ 2 ) 3
  • B 3 π‘₯ βˆ’ 2
  • C 3 ( π‘₯ βˆ’ 2 ) π‘₯
  • D 3 9 π‘₯
  • E 3 π‘₯

Q8:

Let β„Ž ( π‘₯ ) = βˆ’ 4 √ π‘₯ βˆ’ 6 2 . Which of the following functions are the correct 𝑓 and 𝑔 , if β„Ž ( π‘₯ ) = ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) ?

  • A 𝑓 ( π‘₯ ) = βˆ’ 4 √ π‘₯ + 6 , 𝑔 ( π‘₯ ) = π‘₯ 2
  • B 𝑓 ( π‘₯ ) = π‘₯ 2 , 𝑔 ( π‘₯ ) = βˆ’ 4 √ π‘₯ βˆ’ 6
  • C 𝑓 ( π‘₯ ) = 4 √ π‘₯ βˆ’ 6 , 𝑔 ( π‘₯ ) = π‘₯ 2
  • D 𝑓 ( π‘₯ ) = βˆ’ 4 √ π‘₯ βˆ’ 6 , 𝑔 ( π‘₯ ) = π‘₯ 2

Q9:

Let 𝑓 be one-to-one and onto, and let 𝑔 be one-to-one. What is the most that can be said about 𝑔 ∘ 𝑓 ?

  • AThe image of 𝑓 and the image of 𝑔 are the same.
  • B 𝑓 and 𝑔 are inverses.
  • C 𝑔 ∘ 𝑓 is onto.
  • D 𝑔 ∘ 𝑓 is one-to-one.

Q10:

An oil spill grows with time such that the shape resulting remains the same but has an increasing diameter 𝑑 . If the area of the spill is given by 𝐴 ( 𝑑 ) as a function of the diameter, and the diameter is given by 𝐷 ( 𝑑 ) as a function of time 𝑑 , what does 𝐷 ( 𝐴 ( 𝑑 ) ) represent?

  • A the area of the spill as a function of the radius
  • B the area of the spill as a function of time
  • C the area of the spill as a function of the diameter
  • D It does not represent anything.
  • Ethe area of the spill multipled by the diameter

Q11:

Let 𝑓 𝐴 β†’ 𝐡 : and 𝑔 𝐡 β†’ 𝐢 : be maps. Which of the following statements is true?

  • AThe domain of 𝑔 ∘ 𝑓 is 𝐡 .
  • B 𝑓 and 𝑔 are inverses.
  • C 𝐴 = 𝐢
  • DIf ( 𝑔 ∘ 𝑓 ) is 𝐢 , then 𝑓 and 𝑔 are both onto.

Q12:

Given that the function 𝑓 ( π‘₯ ) = 8 π‘₯ + 3 , the function 𝑔 ( π‘₯ ) = π‘₯ + 2 2 , and the function β„Ž ( π‘₯ ) = π‘₯ 3 , determine ( 𝑓 ∘ 𝑔 ) ( βˆ’ 3 ) , ( 𝑔 ∘ β„Ž ) ( 4 ) , and ( β„Ž ∘ 𝑓 ) ( βˆ’ 1 ) .

  • A ( 𝑓 ∘ 𝑔 ) ( βˆ’ 3 ) = 4 4 3 , ( 𝑔 ∘ β„Ž ) ( 4 ) = 4 0 9 8 , ( β„Ž ∘ 𝑓 ) ( βˆ’ 1 ) = βˆ’ 1 2 5
  • B ( 𝑓 ∘ 𝑔 ) ( βˆ’ 3 ) = 4 0 9 8 , ( 𝑔 ∘ β„Ž ) ( 4 ) = 9 1 , ( β„Ž ∘ 𝑓 ) ( βˆ’ 1 ) = βˆ’ 1 2 5
  • C ( 𝑓 ∘ 𝑔 ) ( βˆ’ 3 ) = 8 5 , ( 𝑔 ∘ β„Ž ) ( 4 ) = 4 0 9 8 , ( β„Ž ∘ 𝑓 ) ( βˆ’ 1 ) = βˆ’ 1 3 3 1
  • D ( 𝑓 ∘ 𝑔 ) ( βˆ’ 3 ) = 9 1 , ( 𝑔 ∘ β„Ž ) ( 4 ) = 4 0 9 8 , ( β„Ž ∘ 𝑓 ) ( βˆ’ 1 ) = βˆ’ 1 2 5

Q13:

If the function , and the function , find the domain of .

  • A
  • B
  • C
  • D
  • E

Q14:

If the function , where , and the function , determine the domain of .

  • A
  • B
  • C
  • D
  • E

Q15:

If , and , find the domain of .

  • A
  • B
  • C
  • D

Q16:

If 𝑓 ( π‘₯ ) = ο„ž π‘₯ + 5 2 3 and 𝑔 ( π‘₯ ) = 2 π‘₯ βˆ’ 5 3 , then 𝑔 ( 𝑓 ( π‘₯ ) ) = .

  • A 5 ( π‘₯ + 5 )
  • B 1 . 5 ( π‘₯ + 5 )
  • C π‘₯ βˆ’ 5 3
  • D π‘₯

Q17:

Given 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 , 𝑔 ( π‘₯ ) = π‘₯  , and β„Ž ( π‘₯ ) = 2 π‘₯ , evaluate 𝑓 ( 𝑔 ( β„Ž ( 3 ) ) ) .

Q18:

If the function and the function , find an expression for in its simplest form and determine its domain.

  • A ,
  • B ,
  • C ,
  • D ,

Q19:

Let 𝐴 = { π‘Ž , 𝑏 , 𝑐 } , 𝐡 = { 𝑑 , 𝑒 , 𝑓 } , and 𝐢 = { β„Ž , 𝑖 , 𝑗 } and suppose that 𝑓 ( π‘Ž ) = 𝑑 , 𝑓 ( 𝑏 ) = 𝑓 , 𝑓 ( 𝑐 ) = 𝑒 , 𝑔 ( 𝑑 ) = 𝑖 , 𝑔 ( 𝑒 ) = 𝑗 , and 𝑔 ( 𝑓 ) = β„Ž . Which of the following represents ( 𝑔 ∘ 𝑓 ) ( π‘₯ ) ?

  • A { β„Ž , 𝑗 , 𝑖 }
  • B { β„Ž , 𝑖 , 𝑗 }
  • C { 𝑗 , 𝑖 , π‘˜ }
  • D { 𝑖 , β„Ž , 𝑗 }

Q20:

Given 𝑓 ( 𝑑 ) = 𝑑 βˆ’ 𝑑   and β„Ž ( π‘₯ ) = 4 π‘₯ + 1 , evaluate 𝑓 ( β„Ž ( 1 ) ) .

  • A25
  • B125
  • C1
  • D100

Q21:

If the function , where , and the function , determine the domain of .

  • A
  • B
  • C
  • D
  • E

Q22:

Suppose that 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 𝑐 π‘₯ + 𝑑 and 𝑔 ( π‘₯ ) = βˆ’ 𝑏 π‘₯ + 𝑑 π‘Ž π‘₯ βˆ’ 𝑐 .

Find 𝑓 ( 𝑔 ( π‘₯ ) ) .

  • A π‘Ž π‘₯ + 𝑏 π‘Ž π‘₯ βˆ’ 𝑐
  • B π‘₯
  • C βˆ’ 𝑏 π‘₯ + 𝑑 𝑐 π‘₯ + 𝑑
  • D 1 π‘₯
  • E 𝑐 π‘₯ + 𝑑 π‘Ž π‘₯ βˆ’ 𝑐

Given that β„Ž ( 𝑓 ( π‘₯ ) ) = 1 π‘₯ , find β„Ž in terms of π‘Ž , 𝑏 , 𝑐 , and 𝑑 .

  • A β„Ž ( π‘₯ ) = 𝑐 π‘₯ βˆ’ π‘Ž βˆ’ 𝑑 π‘₯ + 𝑏
  • B β„Ž ( π‘₯ ) = 𝑐 π‘₯ βˆ’ 𝑑 βˆ’ 𝑑 π‘₯ + 𝑏
  • C β„Ž ( π‘₯ ) = 𝑐 π‘₯ βˆ’ π‘Ž βˆ’ 𝑑 π‘₯ + 𝑐
  • D β„Ž ( π‘₯ ) = 𝑏 π‘₯ βˆ’ π‘Ž βˆ’ 𝑐 π‘₯ + 𝑏
  • E β„Ž ( π‘₯ ) = π‘₯ βˆ’ π‘Ž βˆ’ 𝑑 π‘₯ + 𝑏

Q23:

Suppose that 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 4 5 π‘₯ + 7 and 𝑔 ( π‘₯ ) = 4 π‘₯ + 7 2 π‘₯ βˆ’ 5 . Find, in the simplest form, 𝑓 ( 𝑔 ( π‘₯ ) ) and 𝑔 ( 𝑓 ( π‘₯ ) ) .

  • A 𝑓 ( 𝑔 ( π‘₯ ) ) = βˆ’ 4 3 π‘₯ βˆ’ 3 3 2 1 π‘₯ + 4 3 , 𝑔 ( 𝑓 ( π‘₯ ) ) = 1 π‘₯
  • B 𝑓 ( 𝑔 ( π‘₯ ) ) = 1 π‘₯ , 𝑔 ( 𝑓 ( π‘₯ ) ) = 1 π‘₯
  • C 𝑓 ( 𝑔 ( π‘₯ ) ) = 2 π‘₯ βˆ’ 4 2 π‘₯ βˆ’ 5 , 𝑔 ( 𝑓 ( π‘₯ ) ) = 5 π‘₯ + 7 4 π‘₯ + 7
  • D 𝑓 ( 𝑔 ( π‘₯ ) ) = 1 π‘₯ , 𝑔 ( 𝑓 ( π‘₯ ) ) = βˆ’ 4 3 π‘₯ βˆ’ 3 3 2 1 π‘₯ + 4 3
  • E 𝑓 ( 𝑔 ( π‘₯ ) ) = π‘₯ , 𝑔 ( 𝑓 ( π‘₯ ) ) = π‘₯

Q24:

If the function 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 9 3 , and the function 𝑔 ( π‘₯ ) = √ π‘₯ βˆ’ 5 1 2 , find ( 𝑔 ∘ 𝑓 ) ( π‘₯ ) in the simplest form.

  • A √ π‘₯ βˆ’ 4 2
  • B √ π‘₯ + 4 2
  • C  √ π‘₯ βˆ’ 5 1 βˆ’ 9 3 2
  • D √ π‘₯ βˆ’ 1 4 4

Q25:

Which of the following pairs of functions 𝑓 , 𝑔 satisfy ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = √ 9 π‘₯ βˆ’ 2 3 2 ?

  • A 𝑓 ( π‘₯ ) = √ π‘₯ + 2 3 , 𝑔 ( π‘₯ ) = 9 π‘₯ 2
  • B 𝑓 ( π‘₯ ) = 9 π‘₯ 2 , 𝑔 ( π‘₯ ) = √ π‘₯ βˆ’ 2 3
  • C 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 2 3 , 𝑔 ( π‘₯ ) = 3 π‘₯ 2
  • D 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 2 3 , 𝑔 ( π‘₯ ) = 9 π‘₯ 2