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).

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

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‘₯)?

  • A 3 π‘₯ 
  • B 9 π‘₯ βˆ’ 6 π‘₯ + 3 
  • C 3 π‘₯ + 2 
  • D 3 π‘₯ + 3 
  • E 9 π‘₯ βˆ’ 6 π‘₯ + 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
  • B π‘₯ < 3
  • C π‘₯ β‰₯ 3
  • D π‘₯ = 3
  • E π‘₯ ≀ 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‘₯)?

  • A π‘₯   
  • B 3 
  • C 3   
  • D ( π‘₯ βˆ’ 2 ) 
  • E 3 βˆ’ 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‘₯ + 4 6 )  
  • B ο€Ί √ π‘₯ + 4 6   
  • C π‘₯ + 4 6
  • D π‘₯ βˆ’ 4 6

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‘₯))?

  • A 𝑏 𝑐
  • B π‘Ž 𝑏
  • C π‘Ž 𝑐
  • D 𝑏 𝑑
  • E π‘Ž 𝑑

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

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

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.

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

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 ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain=β„βˆ’{βˆ’28,βˆ’5,5}
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain=β„βˆ’{βˆ’5,5}
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 8 1  , domain=β„βˆ’{βˆ’28,βˆ’9,9}
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain=(βˆ’5,5)
  • E ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = βˆ’ 8 π‘₯ βˆ’ 2 5  , 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 ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 8 0 1 , domain=(βˆ’94,∞), (π‘“βˆ˜π‘”)(6)=849
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 7 0 3 , domain=β„βˆ’ο¬βˆ’7038, (π‘“βˆ˜π‘”)(6)=709
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 7 0 3 , domain=[βˆ’94,∞), (π‘“βˆ˜π‘”)(6)=751
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 7 0 3 , domain=β„βˆ’ο¬7038, (π‘“βˆ˜π‘”)(6)=βˆ’655
  • E ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 8 0 1 , 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‘”.

  • A ο€Ό βˆ’ ∞ , 2 5 2 1 9  βˆ’ { 1 3 }
  • B ο€Ό βˆ’ 1 3 , βˆ’ 2 4 2 1 9 
  • C ο€Ό βˆ’ ∞ , βˆ’ 2 4 2 1 9 
  • D  βˆ’ 1 3 , βˆ’ 2 4 2 1 9 
  • E  βˆ’ 2 5 2 1 9 , ∞  βˆ’ { βˆ’ 1 3 }

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 [ 4 1 , ∞ )
  • B ℝ βˆ’ { 4 1 }
  • C ( 4 1 , ∞ )
  • D ℝ βˆ’ { βˆ’ 4 1 }
  • E ℝ βˆ’ { 0 , 4 1 }

Q20:

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

  • A [ 9 1 , ∞ )
  • B ℝ βˆ’ { 9 1 }
  • C ℝ βˆ’ { βˆ’ 9 1 }
  • D ( 9 1 , ∞ )

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 ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) =  √ βˆ’ 1 8 βˆ’ π‘₯ βˆ’ 3 , π‘₯ ∈ ( βˆ’ ∞ , βˆ’ 2 7 ]
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) =  √ 1 8 βˆ’ π‘₯ βˆ’ 3 , π‘₯ ∈ ( βˆ’ ∞ , 9 ]
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) =  1 8 βˆ’ √ π‘₯ βˆ’ 3 , π‘₯ ∈ [ 3 , 3 2 7 ]
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) =  √ 1 8 βˆ’ π‘₯ + 3 , π‘₯ ∈ ( βˆ’ ∞ , 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‘₯).

  • A ℝ βˆ’ { βˆ’ 1 9 , 1 9 }
  • B [ βˆ’ 1 9 , ∞ )
  • C [ 1 9 , ∞ )
  • D ( βˆ’ 1 9 , ∞ )
  • E ℝ βˆ’ { βˆ’ 1 9 , 0 , 1 9 }

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‘“?

  • A 𝑔 ∘ 𝑓 is one-to-one.
  • B 𝑓 and 𝑔 are inverses.
  • C 𝑔 ∘ 𝑓 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?

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

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