Lesson: Composite Functions

In this lesson, we will learn how to form a composite function by composing two or more linear, quadratic, exponential, or radical functions.

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Worksheet: Composite Functions • 25 Questions • 4 Videos

Q1:

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 ) .

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‘₯ ) ?

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‘₯ ?

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‘₯ ) ?

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‘₯ 5 and 𝑔 ( π‘₯ ) = ( π‘₯ + 4 6 ) 5 , find and simplify an expression for ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) .

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?

Q11:

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

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

Q14:

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 ) .

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 .

Q16:

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

Q17:

Given that the function 𝑓 ( π‘₯ ) = 8 π‘₯ βˆ’ 4 9 2 , 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 ) .

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 .

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 .

Q20:

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

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.

Q22:

If the function 𝑓 ( π‘₯ ) = 1 7 π‘₯ , where π‘₯ β‰  0 , and the function 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 3 6 1 2 , determine the domain of ( 𝑓 ∘ 𝑔 ) ( π‘₯ ) .

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 ?

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?

Q25:

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

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