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Lesson: Combining Functions

Sample Question Videos

Worksheet • 25 Questions • 2 Videos

Q1:

Determine the common domain of the functions 𝑛 ( π‘₯ ) = βˆ’ 7 π‘₯ βˆ’ 7 1 and 𝑛 ( π‘₯ ) = βˆ’ 8 π‘₯ βˆ’ 6 4 2 2 .

  • A ℝ βˆ’ { βˆ’ 8 , 7 , 8 }
  • B ℝ βˆ’ { βˆ’ 8 , βˆ’ 7 }
  • C ℝ βˆ’ { βˆ’ 8 , 8 }
  • D ℝ βˆ’ { 7 , 8 }
  • E ℝ βˆ’ { βˆ’ 8 , βˆ’ 7 , 8 }

Q2:

If and are two real functions where and , determine the value of if possible.

  • Aundefined
  • B
  • C
  • D

Q3:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ 2 and 𝑔 ( π‘₯ ) = √ π‘₯ + 1 , find the domain of the function ( 𝑓 + 𝑔 ) .

  • A [ βˆ’ 1 , ∞ [
  • B [ 1 , ∞ [
  • C ] βˆ’ ∞ , βˆ’ 1 ]
  • D ℝ βˆ’ { 0 , 5 }
  • E [ βˆ’ 1 , ∞ [ βˆ’ { 0 , 5 }

Q4:

If 𝑓 ∢ ] βˆ’ 7 , 8 ] β†’ ℝ 1 where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 1 , and 𝑓 ∢ [ βˆ’ 8 , 4 ] β†’ ℝ 2 where 𝑓 ( π‘₯ ) = 4 π‘₯ + 8 π‘₯ + 3 2 2 , find ( 𝑓 βˆ’ 𝑓 ) ( π‘₯ ) 2 1 and the domain of ( 𝑓 βˆ’ 𝑓 ) 2 1 .

  • A 4 π‘₯ + 7 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ 7 , 4 ]
  • B βˆ’ 4 π‘₯ βˆ’ 7 π‘₯ βˆ’ 5 2 , π‘₯ ∈ ] βˆ’ 7 , 4 ]
  • C 4 π‘₯ + 7 π‘₯ + 5 2 , π‘₯ ∈ [ βˆ’ 8 , 4 ]
  • D 4 π‘₯ + 7 π‘₯ + 5 2 , π‘₯ ∈ [ βˆ’ 7 , 4 [
  • E 4 π‘₯ + 7 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ 7 , 8 ]

Q5:

If 𝑓 ∢ ℝ ⟢ ℝ + where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 7 , and 𝑔 ∢ [ βˆ’ 2 5 , 4 ] ⟢ ℝ where 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 1 1 , then find ( 𝑓 + 𝑔 ) ( π‘₯ ) and its domain.

  • A ( 𝑓 + 𝑔 ) ( π‘₯ ) = 2 π‘₯ βˆ’ 2 8 , ] 0 , 4 ]
  • B ( 𝑓 + 𝑔 ) ( π‘₯ ) = 2 π‘₯ βˆ’ 1 7 , [ 0 , 4 ]
  • C ( 𝑓 + 𝑔 ) ( π‘₯ ) = 2 π‘₯ βˆ’ 2 8 , [ 0 , 4 ]
  • D ( 𝑓 + 𝑔 ) ( π‘₯ ) = 2 π‘₯ βˆ’ 1 1 , ] 0 , 4 ]

Q6:

If 𝑓 ℝ β†’ ℝ 1 βˆ’ : where 𝑓 ( π‘₯ ) = 4 π‘₯ + 4 1 , and 𝑓 ] βˆ’ 9 , 6 ] β†’ ℝ 2 : where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 2 , find and fully simplify ( 𝑓 βˆ’ 𝑓 ) ( π‘₯ ) 1 2 and the domain of ( 𝑓 βˆ’ 𝑓 ) 1 2 .

  • A 3 π‘₯ + 5 , π‘₯ ∈ ] βˆ’ 9 , 0 [
  • B 3 π‘₯ + 5 , π‘₯ ∈ ] βˆ’ ∞ , 6 ]
  • C 3 π‘₯ + 5 , π‘₯ ∈ ] βˆ’ 9 , 6 ]
  • D 3 π‘₯ + 5 , π‘₯ ∈ [ βˆ’ 9 , 0 ]
  • E 3 π‘₯ + 5 , π‘₯ ∈ ℝ βˆ’

Q7:

If and are two real functions where and , determine the value of if possible.

  • Aundefined
  • B
  • C1
  • D

Q8:

If and are two real functions where and , determine the value of if possible.

  • Aundefined
  • B5
  • C3
  • D

Q9:

Given that 𝑓 ∢ ℝ β†’ ℝ + , where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 9 , and 𝑔 ∢ [ βˆ’ 2 , 1 3 ] β†’ ℝ , where 𝑔 ( π‘₯ ) = π‘₯ βˆ’ 6 , evaluate ( 𝑓 β‹… 𝑔 ) ( 7 ) .

  • A βˆ’ 1 2
  • B724
  • C βˆ’ 7 7 4
  • D βˆ’ 2 4 0

Q10:

If 𝑓 ℝ β†’ ℝ 1 βˆ’ : where 𝑓 ( π‘₯ ) = βˆ’ π‘₯ βˆ’ 1 1 , and 𝑓 ] βˆ’ 9 , 1 [ β†’ ℝ 2 : where 𝑓 ( π‘₯ ) = 5 π‘₯ βˆ’ 3 2 , find ( 𝑓 + 𝑓 ) ( π‘₯ ) 1 2 and the domain of ( 𝑓 + 𝑓 ) 1 2 .

  • A 4 π‘₯ βˆ’ 4 , π‘₯ ∈ ] βˆ’ 9 , 0 [
  • B 4 π‘₯ βˆ’ 4 , π‘₯ ∈ ] βˆ’ ∞ , 1 [
  • C 4 π‘₯ βˆ’ 4 , π‘₯ ∈ ] βˆ’ 9 , 1 [
  • D 4 π‘₯ βˆ’ 4 , π‘₯ ∈ [ βˆ’ 9 , 0 ]
  • E 4 π‘₯ βˆ’ 4 , π‘₯ ∈ ℝ βˆ’

Q11:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ 2 and 𝑔 ( π‘₯ ) = √ π‘₯ + 4 , determine the domain of the function ( 𝑓 β‹… 𝑔 ) .

  • A [ βˆ’ 4 , ∞ [
  • B [ 4 , ∞ [
  • C ] βˆ’ ∞ , βˆ’ 4 ]
  • D ℝ βˆ’ { 0 , 5 }
  • E [ βˆ’ 4 , ∞ [ βˆ’ { 0 , 5 }

Q12:

Given that and find ο€½ 𝑓 𝑓  ( π‘₯ ) 2 1 and state its domain.

  • A 2 π‘₯ + 3 , π‘₯ ∈ ] βˆ’ 8 , 4 [ βˆ’ { βˆ’ 5 }
  • B 2 π‘₯ + 3 , π‘₯ ∈ ] βˆ’ ∞ , 4 [ βˆ’ { βˆ’ 5 }
  • C 2 π‘₯ + 3 , π‘₯ ∈ ] βˆ’ ∞ , 4 [
  • D 2 π‘₯ + 3 , π‘₯ ∈ ] βˆ’ 8 , 4 [
  • E 2 π‘₯ + 3 , π‘₯ ∈ ] βˆ’ 8 , 6 [

Q13:

Given that and find ο€½ 𝑓 𝑓  ( π‘₯ ) 2 1 and state its domain.

  • A π‘₯ + 1 , π‘₯ ∈ ] βˆ’ 6 , 0 ] βˆ’ { βˆ’ 5 }
  • B π‘₯ + 1 , π‘₯ ∈ [ βˆ’ 7 , ∞ [ βˆ’ { βˆ’ 5 }
  • C π‘₯ + 1 , π‘₯ ∈ [ βˆ’ 7 , ∞ [
  • D π‘₯ + 1 , π‘₯ ∈ [ βˆ’ 6 , 0 [
  • E π‘₯ + 1 , π‘₯ ∈ ] βˆ’ 6 , 0 ]

Q14:

If 𝑓 ℝ β†’ ℝ : where 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 4 , and 𝑔 [ βˆ’ 8 , βˆ’ 2 [ β†’ ℝ : where 𝑔 ( π‘₯ ) = 5 π‘₯ + 5 , find the value of ( 𝑓 + 𝑔 ) ( 5 ) if possible.

  • Aundefined
  • B46
  • C36
  • D16

Q15:

What is the domain of the quotient 𝑓 𝑔 , in terms of the domains of 𝑓 and 𝑔 ? Assume that both domains are subsets of the set of real numbers.

  • A the intersection of the domain of 𝑓 and the domain of 1 𝑔
  • Bthe difference between the domain of 𝑓 and the domain of 𝑔
  • C the intersection of the domain of 𝑓 and the domain of 𝑔
  • D the larger of the domain of 𝑓 and the domain of 𝑔
  • E the union of the domain of 𝑓 and the domain of 𝑔

Q16:

Given that and find ( 𝑓 β‹… 𝑓 ) ( π‘₯ ) 1 2 and state its domain.

  • A 5 π‘₯ βˆ’ 2 2 π‘₯ + 8 2 , π‘₯ ∈ ] 0 , 1 ]
  • B 5 π‘₯ βˆ’ 2 2 π‘₯ + 8 2 , π‘₯ ∈ ] βˆ’ 9 , ∞ [
  • C 5 π‘₯ βˆ’ 2 2 π‘₯ + 8 2 , π‘₯ ∈ ] βˆ’ 9 , 1 ]
  • D 5 π‘₯ βˆ’ 2 2 π‘₯ + 8 2 , π‘₯ ∈ [ 0 , 1 [
  • E 5 π‘₯ βˆ’ 2 2 π‘₯ + 8 2 , π‘₯ ∈ ℝ +

Q17:

If 𝑓 and 𝑔 are two real functions where and 𝑔 ( π‘₯ ) = π‘₯ , find the domain of the function ( 𝑓 β‹… 𝑔 ) .

  • A ] 0 , ∞ [
  • B ℝ
  • C ] 0 , 2 [
  • D ] 0 , ∞ [ βˆ’ { 2 }
  • E [ 2 , ∞ [

Q18:

Given that and find ο€½ 𝑓 𝑓  ( π‘₯ ) 2 1 and state its domain.

  • A 2 π‘₯ βˆ’ π‘₯ βˆ’ 6 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ ∞ , βˆ’ 1 [ βˆ’ { βˆ’ 5 }
  • B π‘₯ + 5 2 π‘₯ βˆ’ π‘₯ βˆ’ 6 2 , π‘₯ ∈ ] βˆ’ ∞ , βˆ’ 1 [ βˆ’ { βˆ’ 5 }
  • C 2 π‘₯ βˆ’ π‘₯ βˆ’ 6 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ ∞ , βˆ’ 1 [
  • D 2 π‘₯ βˆ’ π‘₯ βˆ’ 6 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ ∞ , 2 ]
  • E 2 π‘₯ βˆ’ π‘₯ βˆ’ 6 π‘₯ + 5 2 , π‘₯ ∈ ] βˆ’ ∞ , βˆ’ 1 ]

Q19:

Given that and find the value of ο€½ 𝑓 𝑔  ( βˆ’ 1 ) if possible.

  • Anot defined
  • B 1 2
  • C0
  • D βˆ’ 1

Q20:

Given that and find ( 𝑓 β‹… 𝑓 ) ( π‘₯ ) 1 2 and state its domain.

  • A π‘₯ + 7 π‘₯ βˆ’ 5 π‘₯ βˆ’ 7 5 3 2 , π‘₯ ∈ ] 3 , 6 ]
  • B π‘₯ + 7 π‘₯ βˆ’ 5 π‘₯ βˆ’ 7 5 3 2 , π‘₯ ∈ ] βˆ’ 1 , 9 [
  • C π‘₯ βˆ’ 3 0 π‘₯ + 1 0 π‘₯ βˆ’ 7 5 3 2 , π‘₯ ∈ ] 3 , 6 ]
  • D π‘₯ + 7 π‘₯ βˆ’ 5 π‘₯ βˆ’ 7 5 3 2 , π‘₯ ∈ [ 3 , 6 [

Q21:

Given that , , and , determine in its simplest form.

  • A
  • B
  • C
  • D
  • E

Q22:

Given that 𝑛 ( π‘₯ ) = 5 π‘₯ βˆ’ 8 2 5 π‘₯ βˆ’ 4 Γ· 2 5 π‘₯ βˆ’ 3 0 π‘₯ βˆ’ 1 6 1 2 5 π‘₯ + 8 1 2 2 3 , 𝑛 ( π‘₯ ) = 2 5 π‘₯ βˆ’ 4 5 0 π‘₯ βˆ’ 2 0 π‘₯ + 8 2 2 2 , and 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ ) Γ— 𝑛 ( π‘₯ ) 1 2 , simplify the function 𝑛 , and determine its domain.

  • A 𝑛 = 1 2 , domain = ℝ βˆ’  βˆ’ 2 5 , 2 5 , 8 5 
  • B 𝑛 = 5 2 , domain = ℝ βˆ’  βˆ’ 2 5 , 2 5 , 8 5 
  • C 𝑛 = 1 2 , domain = ℝ βˆ’  βˆ’ 2 5 , 2 5 
  • D 𝑛 = 2 , domain = ℝ βˆ’  βˆ’ 2 5 , 2 5 
  • E 𝑛 = 2 , domain = ℝ βˆ’  βˆ’ 2 5 , 2 5 , 8 5 

Q23:

Given that 𝑛 ( π‘₯ ) = π‘₯ βˆ’ 3 6 4 π‘₯ βˆ’ 1 Γ· 8 π‘₯ βˆ’ 2 3 π‘₯ βˆ’ 3 5 1 2 π‘₯ + 1 1 2 2 3 , 𝑛 ( π‘₯ ) = 2 5 6 π‘₯ βˆ’ 4 3 2 0 π‘₯ βˆ’ 4 0 π‘₯ + 5 2 2 2 , and 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ ) Γ— 𝑛 ( π‘₯ ) 1 2 , simplify the function 𝑛 , and determine its domain.

  • A 𝑛 = 4 5 , domain = ℝ βˆ’  βˆ’ 1 8 , 1 8 , 3 
  • B 𝑛 = 8 5 , domain = ℝ βˆ’  βˆ’ 1 8 , 1 8 , 3 
  • C 𝑛 = 4 5 , domain = ℝ βˆ’  βˆ’ 1 8 , 1 8 
  • D 𝑛 = 5 4 , domain = ℝ βˆ’  βˆ’ 1 8 , 1 8 
  • E 𝑛 = 5 4 , domain = ℝ βˆ’  βˆ’ 1 8 , 1 8 , 3 

Q24:

If 𝑓 ( π‘₯ ) = π‘₯ + 1 and 𝑔 ( π‘₯ ) = π‘₯ + 1 2 , then find and fully simplify an expression for ( 𝑓 β‹… 𝑔 ) ( π‘₯ ) .

  • A π‘₯ + π‘₯ + π‘₯ + 1 3 2
  • B π‘₯ + π‘₯ + 2 2
  • C π‘₯ + π‘₯ + 1 3
  • D π‘₯ + π‘₯ + 1 3 2

Q25:

If 𝑓 and 𝑔 are two real functions where and 𝑔 ( π‘₯ ) = 5 π‘₯ determine the domain of the function ο€½ 𝑔 𝑓  .

  • A ] βˆ’ ∞ , 0 [
  • B ] βˆ’ ∞ , βˆ’ 3 [
  • C ℝ βˆ’ { 0 }
  • D [ βˆ’ 3 , 0 [
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