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(𝑥)=19𝑥 and the function 𝑔(𝑥)=2𝑥, determine (𝑔𝑓)(𝑥) in its simplest form, and evaluate (𝑔𝑓)(1).

  • A38𝑥, (𝑔𝑓)(1)=38
  • B38𝑥, (𝑔𝑓)(1)=38
  • C19𝑥, (𝑔𝑓)(1)=19
  • D76𝑥, (𝑔𝑓)(1)=76

Q2:

If 𝑓(𝑥)=3𝑥 and 𝑔(𝑥)=2𝑥+4, find (𝑓𝑔)(1).

Q3:

Given 𝑓(𝑥)=3𝑥1 and 𝑔(𝑥)=𝑥+1, which of the following expressions gives (𝑓𝑔)(𝑥)?

  • A3𝑥
  • B9𝑥6𝑥+3
  • C3𝑥+2
  • D3𝑥+3
  • E9𝑥6𝑥+2

Q4:

Given 𝑓(𝑥)=3𝑥1 and 𝑔(𝑥)=𝑥+1, find (𝑓𝑔)(2).

Q5:

Let 𝑓(𝑥)=2|𝑥3|4 and 𝑔(𝑥)=2𝑥2. For what values of 𝑥 is it true that 𝑔(𝑓(𝑥))=𝑥?

  • Aall real numbers
  • B𝑥<3
  • C𝑥3
  • D𝑥=3
  • E𝑥3

Q6:

If 𝑓(𝑥)=3𝑥 and 𝑔(𝑥)=2𝑥+4, find 𝑓(𝑔(1)).

Q7:

If 𝑓(𝑥)=3 and 𝑔(𝑥)=𝑥2, what is (𝑓𝑔)(𝑥)?

  • A𝑥
  • B3
  • C3
  • D(𝑥2)
  • E32

Q8:

Given that 𝑓(𝑥)=3𝑥+2, find𝐵 so that 𝑔(𝑥)=3𝑥+𝐵 satisfies 𝑓𝑔=𝑔𝑓.

Q9:

Given that 𝑓(𝑥)=𝑥 and 𝑔(𝑥)=(𝑥+46), find and simplify an expression for (𝑓𝑔)(𝑥).

  • A(𝑥+46)
  • B𝑥+46
  • C𝑥+46
  • D𝑥46

Q10:

If 𝑓(𝑥)=𝑎𝑥+𝑏 and 𝑔(𝑥)=𝑐𝑥+𝑑, what is the coefficient of 𝑥 in 𝑓(𝑔(𝑥))?

  • A𝑏𝑐
  • B𝑎𝑏
  • C𝑎𝑐
  • D𝑏𝑑
  • E𝑎𝑑

Q11:

In the given figure, the red graph represents 𝑦=𝑓(𝑥), while the blue represents 𝑦=𝑔(𝑥).

What is 𝑓(𝑔(2))?

Q12:

Given that the function 𝑓(𝑥)=8𝑥+3, the function 𝑔(𝑥)=𝑥+2, and the function (𝑥)=𝑥, determine (𝑓𝑔)(3), (𝑔)(4), and (𝑓)(1).

  • A(𝑓𝑔)(3)=91, (𝑔)(4)=4,098, (𝑓)(1)=125
  • B(𝑓𝑔)(3)=4,098, (𝑔)(4)=91, (𝑓)(1)=125
  • C(𝑓𝑔)(3)=85, (𝑔)(4)=4,098, (𝑓)(1)=1,331
  • D(𝑓𝑔)(3)=443, (𝑔)(4)=4,098, (𝑓)(1)=125

Q13:

Given that the function 𝑓(𝑥)=𝑥89, and the function 𝑔(𝑥)=𝑥+17, find (𝑓𝑔)(𝑥) in its simplest form, then determine (𝑓𝑔)(19).

  • A(𝑓𝑔)(𝑥)=𝑥72, (𝑓𝑔)(19)=53
  • B(𝑓𝑔)(𝑥)=𝑥+106, (𝑓𝑔)(19)=125
  • C(𝑓𝑔)(𝑥)=𝑥+1789, (𝑓𝑔)(19)=83
  • D(𝑓𝑔)(𝑥)=𝑥106, (𝑓𝑔)(19)=87
  • E(𝑓𝑔)(𝑥)=𝑥72, (𝑓𝑔)(19)=17

Q14:

For 𝑓(𝑥)=3 and 𝑔(𝑥)=𝑥2, express (𝑓𝑔)(𝑥) in the form 𝐴𝑏 with suitable numbers for 𝐴 and 𝑏.

  • A3
  • B3
  • C(𝑥2)
  • D3(𝑥2)
  • E39

Q15:

Given that the function 𝑓(𝑥)=8𝑥+28, and the function 𝑔(𝑥)=𝑥53, determine (𝑓𝑔)(𝑥) in its simplest form, and find its domain.

  • A(𝑓𝑔)(𝑥)=8𝑥25, domain={28,5,5}
  • B(𝑓𝑔)(𝑥)=8𝑥25, domain={5,5}
  • C(𝑓𝑔)(𝑥)=8𝑥81, domain={28,9,9}
  • D(𝑓𝑔)(𝑥)=8𝑥25, domain=(5,5)
  • E(𝑓𝑔)(𝑥)=8𝑥25, domain={5,5}

Q16:

Given that the function 𝑓(𝑥)=8𝑥49, and the function 𝑔(𝑥)=𝑥+94, express (𝑓𝑔)(𝑥) in its simplest form, and find its domain, then evaluate (𝑓𝑔)(6).

  • A(𝑓𝑔)(𝑥)=8𝑥+801, domain=(94,), (𝑓𝑔)(6)=849
  • B(𝑓𝑔)(𝑥)=8𝑥+703, domain=7038, (𝑓𝑔)(6)=709
  • C(𝑓𝑔)(𝑥)=8𝑥+703, domain=[94,), (𝑓𝑔)(6)=751
  • D(𝑓𝑔)(𝑥)=8𝑥703, domain=7038, (𝑓𝑔)(6)=655
  • E(𝑓𝑔)(𝑥)=8𝑥801, domain=, (𝑓𝑔)(6)=753

Q17:

If the function 𝑓(𝑥)=𝑥19, and the function 𝑔(𝑥)=5𝑥+13, find the domain of 𝑓𝑔.

  • A,25219{13}
  • B13,24219
  • C,24219
  • D13,24219
  • E25219,{13}

Q18:

If the function 𝑓(𝑥)=2𝑥, where 𝑥0, and the function 𝑔(𝑥)=𝑥41, determine the domain of 𝑓𝑔.

  • A[41,)
  • B{41}
  • C(41,)
  • D{41}
  • E{0,41}

Q19:

If 𝑓(𝑥)=4𝑥91, and 𝑔(𝑥)=𝑥+55, find the domain of 𝑔𝑓.

  • A[91,)
  • B{91}
  • C{91}
  • D(91,)

Q20:

If the function 𝑓(𝑥)=𝑥3 and the function 𝑔(𝑥)=18𝑥, find an expression for (𝑓𝑔)(𝑥) in its simplest form and determine its domain.

  • A(𝑓𝑔)(𝑥)=18𝑥3, 𝑥(,27]
  • B(𝑓𝑔)(𝑥)=18𝑥3, 𝑥(,9]
  • C(𝑓𝑔)(𝑥)=18𝑥3, 𝑥[3,327]
  • D(𝑓𝑔)(𝑥)=18𝑥+3, 𝑥(,9]

Q21:

If the function 𝑓(𝑥)=17𝑥, where 𝑥0, and the function 𝑔(𝑥)=𝑥361, determine the domain of (𝑓𝑔)(𝑥).

  • A{19,19}
  • B[19,)
  • C[19,)
  • D(19,)
  • E{19,0,19}

Q22:

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(𝑡) as a function of time 𝑡, what does 𝐷(𝐴(𝑡)) 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

Q23:

Suppose 𝑓(𝑥)=3 and 𝐿(𝑥)=𝑚𝑥+𝑐. If (𝑓𝐿)(𝑥) is expressed in the form 𝐴𝑏, determine the numbers 𝐴 and 𝑏.

  • A𝐴=3, 𝑏=3
  • B𝐴=3, 𝑏=3
  • C𝐴=3, 𝑏=𝑐𝑚
  • D𝐴=3, 𝑏=3
  • E𝐴=3𝑐, 𝑏=𝑚

Q24:

Let (𝑥)=4𝑥6. Which of the following functions are the correct 𝑓 and 𝑔, if (𝑥)=(𝑓𝑔)(𝑥)?

  • A𝑓(𝑥)=4𝑥6, 𝑔(𝑥)=𝑥
  • B𝑓(𝑥)=𝑥, 𝑔(𝑥)=4𝑥6
  • C𝑓(𝑥)=4𝑥+6, 𝑔(𝑥)=𝑥
  • D𝑓(𝑥)=4𝑥6, 𝑔(𝑥)=𝑥

Q25:

If the function 𝑓(𝑥)=𝑥93, and the function 𝑔(𝑥)=𝑥51, find (𝑔𝑓)(𝑥) in the simplest form.

  • A𝑥42
  • B𝑥5193
  • C𝑥144
  • D𝑥+42

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