Worksheet: Composite Functions

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

Q1:

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

Q2:

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

Q3:

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

Q4:

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

Q5:

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

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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

Q11:

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‘‘

Q12:

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

What is 𝑓(𝑔(2))?

Q13:

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

Q14:

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

Q15:

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

Q16:

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}

Q17:

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

Q18:

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}

Q19:

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}

Q20:

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,∞)

Q21:

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]

Q22:

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}

Q23:

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.

Q24:

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

Q25:

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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