Worksheet: Composite Functions

In this worksheet, we will practice forming a composite function by composing two or more linear, quadratic, exponential, or radical functions.


Given that the function 𝑓(π‘₯)=19π‘₯ and the function 𝑔(π‘₯)=βˆ’2π‘₯, determine (π‘”βˆ˜π‘“)(π‘₯) in its simplest form, and evaluate (π‘”βˆ˜π‘“)(1).

  • A38π‘₯, (π‘”βˆ˜π‘“)(1)=38
  • Bβˆ’38π‘₯, (π‘”βˆ˜π‘“)(1)=βˆ’38
  • C19π‘₯, (π‘”βˆ˜π‘“)(1)=19
  • D76π‘₯, (π‘”βˆ˜π‘“)(1)=76


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


Given 𝑓(π‘₯)=3π‘₯βˆ’1 and 𝑔(π‘₯)=π‘₯+1, which of the following expressions gives (π‘“βˆ˜π‘”)(π‘₯)?

  • A3π‘₯
  • B9π‘₯βˆ’6π‘₯+3
  • C3π‘₯+2
  • D3π‘₯+3
  • E9π‘₯βˆ’6π‘₯+2


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


Let 𝑓(π‘₯)=2|π‘₯βˆ’3|βˆ’4 and 𝑔(π‘₯)=2βˆ’π‘₯2. For what values of π‘₯ is it true that 𝑔(𝑓(π‘₯))=π‘₯?

  • Aall real numbers
  • Bπ‘₯<3
  • Cπ‘₯β‰₯3
  • Dπ‘₯=3
  • Eπ‘₯≀3


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


If 𝑓(π‘₯)=3 and 𝑔(π‘₯)=π‘₯βˆ’2, what is (π‘“βˆ˜π‘”)(π‘₯)?

  • Aπ‘₯ο—οŠ±οŠ¨
  • B3
  • C3ο—οŠ±οŠ¨
  • D(π‘₯βˆ’2)
  • E3βˆ’2


Given that 𝑓(π‘₯)=3π‘₯+2, find𝐡 so that 𝑔(π‘₯)=βˆ’3π‘₯+𝐡 satisfies π‘“βˆ˜π‘”=π‘”βˆ˜π‘“.


Given that 𝑓(π‘₯)=√π‘₯ and 𝑔(π‘₯)=(π‘₯+46), find and simplify an expression for (π‘“βˆ˜π‘”)(π‘₯).

  • A(π‘₯+46)
  • Bο€Ίβˆšπ‘₯+46ο†οŽ€οŠ«
  • Cπ‘₯+46
  • Dπ‘₯βˆ’46


The function 𝐴(𝑑) gives the pain level on a scale of 0 to 10 experienced by a patient with 𝑑 milligrams of a pain-reducing drug in their system. The number of milligrams of the drug in the patient’s system after 𝑑 minutes is modeled by π‘š(𝑑). Which of the following would you do in order to determine when the patient will be at a pain level of 4?

  • AEvaluating π‘š(𝐴(4))
  • BEvaluating 𝐴(π‘š(4))
  • CSolving π‘š(𝐴(𝑑))=4
  • DSolving 𝐴(π‘š(𝑑))=4


If 𝑓(π‘₯)=π‘Žπ‘₯+𝑏 and 𝑔(π‘₯)=𝑐π‘₯+𝑑, what is the coefficient of π‘₯ in 𝑓(𝑔(π‘₯))?

  • A𝑏𝑐
  • Bπ‘Žπ‘
  • Cπ‘Žπ‘
  • D𝑏𝑑
  • Eπ‘Žπ‘‘


In the given figure, the red graph represents 𝑦=𝑓(π‘₯), while the blue represents 𝑦=𝑔(π‘₯).

What is 𝑓(𝑔(2))?


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

  • A(π‘“βˆ˜π‘”)(βˆ’3)=91, (π‘”βˆ˜β„Ž)(4)=4,098, (β„Žβˆ˜π‘“)(βˆ’1)=βˆ’125
  • B(π‘“βˆ˜π‘”)(βˆ’3)=4,098, (π‘”βˆ˜β„Ž)(4)=91, (β„Žβˆ˜π‘“)(βˆ’1)=βˆ’125
  • C(π‘“βˆ˜π‘”)(βˆ’3)=85, (π‘”βˆ˜β„Ž)(4)=4,098, (β„Žβˆ˜π‘“)(βˆ’1)=βˆ’1,331
  • D(π‘“βˆ˜π‘”)(βˆ’3)=443, (π‘”βˆ˜β„Ž)(4)=4,098, (β„Žβˆ˜π‘“)(βˆ’1)=βˆ’125


Given that the function 𝑓(π‘₯)=π‘₯βˆ’89, and the function 𝑔(π‘₯)=√π‘₯+17, find (π‘“βˆ˜π‘”)(π‘₯) in its simplest form, then determine (π‘“βˆ˜π‘”)(19).

  • A(π‘“βˆ˜π‘”)(π‘₯)=π‘₯βˆ’72, (π‘“βˆ˜π‘”)(19)=βˆ’53
  • B(π‘“βˆ˜π‘”)(π‘₯)=π‘₯+106, (π‘“βˆ˜π‘”)(19)=125
  • C(π‘“βˆ˜π‘”)(π‘₯)=ο€»βˆšπ‘₯+17ο‡βˆ’89, (π‘“βˆ˜π‘”)(19)=βˆ’83
  • D(π‘“βˆ˜π‘”)(π‘₯)=π‘₯βˆ’106, (π‘“βˆ˜π‘”)(19)=βˆ’87
  • E(π‘“βˆ˜π‘”)(π‘₯)=√π‘₯βˆ’72, (π‘“βˆ˜π‘”)(19)=17


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

  • A3
  • B3ο—οŠ±οŠ¨
  • C(π‘₯βˆ’2)
  • D3(π‘₯βˆ’2)
  • E39


Given that the function 𝑓(π‘₯)=8π‘₯+28, and the function 𝑔(π‘₯)=π‘₯βˆ’53, determine (π‘“βˆ˜π‘”)(π‘₯) in its simplest form, and find its domain.

  • A(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’25, domain=β„βˆ’{βˆ’28,βˆ’5,5}
  • B(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’25, domain=β„βˆ’{βˆ’5,5}
  • C(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’81, domain=β„βˆ’{βˆ’28,βˆ’9,9}
  • D(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’25, domain=(βˆ’5,5)
  • E(π‘“βˆ˜π‘”)(π‘₯)=βˆ’8π‘₯βˆ’25, domain=β„βˆ’{βˆ’5,5}


Given that the function 𝑓(π‘₯)=8π‘₯βˆ’49, and the function 𝑔(π‘₯)=√π‘₯+94, express (π‘“βˆ˜π‘”)(π‘₯) in its simplest form, and find its domain, then evaluate (π‘“βˆ˜π‘”)(6).

  • A(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯+801, domain=(βˆ’94,∞), (π‘“βˆ˜π‘”)(6)=849
  • B(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯+703, domain=β„βˆ’ο¬βˆ’7038, (π‘“βˆ˜π‘”)(6)=709
  • C(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯+703, domain=[βˆ’94,∞), (π‘“βˆ˜π‘”)(6)=751
  • D(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’703, domain=β„βˆ’ο¬7038, (π‘“βˆ˜π‘”)(6)=βˆ’655
  • E(π‘“βˆ˜π‘”)(π‘₯)=8π‘₯βˆ’801, domain=ℝ, (π‘“βˆ˜π‘”)(6)=βˆ’753


If the function 𝑓(π‘₯)=√π‘₯βˆ’19, and the function 𝑔(π‘₯)=5π‘₯+13, find the domain of π‘“βˆ˜π‘”.

  • Aο€Όβˆ’βˆž,25219ο βˆ’{13}
  • Bο€Όβˆ’13,βˆ’24219
  • Cο€Όβˆ’βˆž,βˆ’24219
  • Dο”βˆ’13,βˆ’24219
  • Eο”βˆ’25219,βˆžοˆβˆ’{βˆ’13}


If the function 𝑓(π‘₯)=2π‘₯, where π‘₯β‰ 0, and the function 𝑔(π‘₯)=π‘₯βˆ’41, determine the domain of π‘“βˆ˜π‘”.

  • A[41,∞)
  • Bβ„βˆ’{41}
  • C(41,∞)
  • Dβ„βˆ’{βˆ’41}
  • Eβ„βˆ’{0,41}


If 𝑓(π‘₯)=βˆ’4π‘₯βˆ’91, and 𝑔(π‘₯)=π‘₯+55, find the domain of π‘”βˆ˜π‘“.

  • A[91,∞)
  • Bβ„βˆ’{91}
  • Cβ„βˆ’{βˆ’91}
  • D(91,∞)


If the function 𝑓(π‘₯)=√π‘₯βˆ’3 and the function 𝑔(π‘₯)=√18βˆ’π‘₯, find an expression for (π‘“βˆ˜π‘”)(π‘₯) in its simplest form and determine its domain.

  • A(π‘“βˆ˜π‘”)(π‘₯)=ο„βˆšβˆ’18βˆ’π‘₯βˆ’3, π‘₯∈(βˆ’βˆž,βˆ’27]
  • B(π‘“βˆ˜π‘”)(π‘₯)=ο„βˆš18βˆ’π‘₯βˆ’3, π‘₯∈(βˆ’βˆž,9]
  • C(π‘“βˆ˜π‘”)(π‘₯)=18βˆ’βˆšπ‘₯βˆ’3, π‘₯∈[3,327]
  • D(π‘“βˆ˜π‘”)(π‘₯)=ο„βˆš18βˆ’π‘₯+3, π‘₯∈(βˆ’βˆž,9]


If the function 𝑓(π‘₯)=17π‘₯, where π‘₯β‰ 0, and the function 𝑔(π‘₯)=π‘₯βˆ’361, determine the domain of (π‘“βˆ˜π‘”)(π‘₯).

  • Aβ„βˆ’{βˆ’19,19}
  • B[βˆ’19,∞)
  • C[19,∞)
  • D(βˆ’19,∞)
  • Eβ„βˆ’{βˆ’19,0,19}


Let 𝑓 be one-to-one and onto, and let 𝑔 be one-to-one. What is the most that can be said about π‘”βˆ˜π‘“?

  • Aπ‘”βˆ˜π‘“ is one-to-one.
  • B𝑓 and 𝑔 are inverses.
  • Cπ‘”βˆ˜π‘“ is onto.
  • DThe image of 𝑓 and the image of 𝑔 are the same.


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?

  • AIt does not represent anything.
  • Bthe area of the spill multipled by the diameter
  • Cthe area of the spill as a function of the radius
  • Dthe area of the spill as a function of time
  • Ethe area of the spill as a function of the diameter


Let 𝑓𝐴→𝐡: and 𝑔𝐡→𝐢: be maps. Which of the following statements is true?

  • A𝑓 and 𝑔 are inverses.
  • BThe domain of π‘”βˆ˜π‘“ is 𝐡.
  • C𝐴=𝐢
  • DIf (π‘”βˆ˜π‘“) is 𝐢, then 𝑓 and 𝑔 are both onto.

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