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

Q2:

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

Q3:

Given 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 1 and 𝑔 ( π‘₯ ) = π‘₯ + 1 2 , which of the following expressions gives ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) ?

  • A 3 π‘₯ 2
  • B 9 π‘₯ βˆ’ 6 π‘₯ + 2 2
  • C 9 π‘₯ βˆ’ 6 π‘₯ + 3 2
  • D 3 π‘₯ + 2 2
  • E 3 π‘₯ + 3 2

Q4:

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

Q5:

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

  • A π‘₯ β‰₯ 3
  • B all real numbers
  • C π‘₯ < 3
  • D π‘₯ ≀ 3
  • E π‘₯ = 3

Q6:

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

Q7:

If 𝑓 ( π‘₯ ) = 3 π‘₯ and 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 2 , what is ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) ?

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

Q8:

Given that 𝑓 ( π‘₯ ) = 3 π‘₯ + 2 , find 𝐡 so that 𝑔 ( π‘₯ ) = βˆ’ 3 π‘₯ + 𝐡 satisfies 𝑓 ∘ 𝑔 = 𝑔 ∘ 𝑓 .

Q9:

Given that 𝑓 ( π‘₯ ) = √ π‘₯  and 𝑔 ( π‘₯ ) = ( π‘₯ + 4 6 )  , find and simplify an expression for ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) .

  • A ( π‘₯ + 4 6 )  
  • B π‘₯ βˆ’ 4 6
  • C ο€Ί √ π‘₯ + 4 6   
  • D π‘₯ + 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 𝐴 ( π‘š ( 4 ) )
  • BEvaluating π‘š ( 𝐴 ( 4 ) )
  • CSolving π‘š ( 𝐴 ( 𝑑 ) ) = 4
  • DSolving 𝐴 ( π‘š ( 𝑑 ) ) = 4

Q11:

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

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

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‘₯ ) = 8 π‘₯ + 3 , the function 𝑔 ( π‘₯ ) = π‘₯  + 2 , and the function β„Ž ( π‘₯ ) = π‘₯  , 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

Q14:

Given that the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 9  , 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  , ( 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

Q15:

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‘₯

Q16:

Given that the function 𝑓 ( π‘₯ ) = 8 π‘₯ + 2 8 , and the function 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 5 3  , determine ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) in its simplest form, and find its domain.

  • A ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 8 1  , domain = ℝ βˆ’ { βˆ’ 2 8 , βˆ’ 9 , 9 }
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain = ℝ βˆ’ { βˆ’ 2 8 , βˆ’ 5 , 5 }
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = βˆ’ 8 π‘₯ βˆ’ 2 5  , domain = ℝ βˆ’ { βˆ’ 5 , 5 }
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain = ℝ βˆ’ { βˆ’ 5 , 5 }
  • E ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 2 5  , domain = ( βˆ’ 5 , 5 )

Q17:

Given that the function 𝑓 ( π‘₯ ) = 8 π‘₯ βˆ’ 4 9  , and the function 𝑔 ( π‘₯ ) = √ π‘₯ + 9 4 , express ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) in its simplest form, and find its domain, then evaluate ( 𝑓 ∘ 𝑔 ) ( 6 ) .

  • A ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 8 0 1 , domain = ℝ , ( 𝑓 ∘ 𝑔 ) ( 6 ) = βˆ’ 7 5 3
  • B ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 8 0 1 , domain = ( βˆ’ 9 4 , ∞ ) , ( 𝑓 ∘ 𝑔 ) ( 6 ) = 8 4 9
  • C ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 7 0 3 , domain = ℝ βˆ’  βˆ’ 7 0 3 8  , ( 𝑓 ∘ 𝑔 ) ( 6 ) = 7 0 9
  • D ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ + 7 0 3 , domain = [ βˆ’ 9 4 , ∞ ) , ( 𝑓 ∘ 𝑔 ) ( 6 ) = 7 5 1
  • E ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) = 8 π‘₯ βˆ’ 7 0 3 , domain = ℝ βˆ’  7 0 3 8  , ( 𝑓 ∘ 𝑔 ) ( 6 ) = βˆ’ 6 5 5

Q18:

If the function 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 1 9 , and the function 𝑔 ( π‘₯ ) = 5 π‘₯ + 1 3 , find the domain of 𝑓 ∘ 𝑔 .

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

Q19:

If the function 𝑓 ( π‘₯ ) = 2 π‘₯ , where π‘₯ β‰  0 , and the function 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 4 1 , determine the domain of 𝑓 ∘ 𝑔 .

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

Q20:

If 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ βˆ’ 9 1 , and 𝑔 ( π‘₯ ) = π‘₯ + 5 5 , find the domain of 𝑔 ∘ 𝑓 .

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

Q21:

If the function 𝑓 ( π‘₯ ) = √ π‘₯ βˆ’ 3 and the function 𝑔 ( π‘₯ ) = √ 1 8 βˆ’ π‘₯ , find an expression for ( 𝑓 ∘ 𝑔 ) ( 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‘₯ + 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‘₯ βˆ’ 3 , π‘₯ ∈ ( βˆ’ ∞ , 9 ]

Q22:

If the function 𝑓 ( π‘₯ ) = 1 7 π‘₯ , where π‘₯ β‰  0 , and the function 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 3 6 1  , determine the domain of ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) .

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

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?

  • 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

Q25:

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.

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