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Functions, 2ADV F1 EQ-Bank 45

Determine whether the function  \(f(x)=2x^3-5x\)  is even, odd or neither. Show all working.   (2 marks)

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\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Show Worked Solution

\(f(x)=2x^{3}-5x\)

\(\text{Function is odd if:}\ \ f(-x)=-f(x) \)

\(f(-x)\) \(=2(-x)^{3}-5(-x) \)  
  \(=-2x^{3}+5x \)  
  \(=-(2x^{3}-5x)\)  
  \(=-f(x)\)  

 
\(\therefore f(x)\ \text{is odd.}\)

Functions, 2ADV F1 2023 HSC 4 MC

The graph of a polynomial is shown.
 


  

Which row of the table is correct for this polynomial?
 

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`B`

Show Worked Solution

`text{By elimination:}`

`text{Double root where graph SP touches}\ xtext{-axis}`

`:.\ C and D\ text{incorrect}`

`(x-c)^2\ text{occurs where}\ x>0`

`:. c>0\ (A\ text{is incorrect})`

`=>B`

Functions, 2ADV F1 2021 HSC 8 MC

The graph of  `y = f(x)`  is shown.

Which of the following could be the equation of this graph?

  1. `y = (1 - x)(2 + x)^3`
  2. `y = (x + 1)(x - 2)^3`
  3. `y = (x + 1)(2 - x)^3`
  4. `y = (x - 1)(2 + x)^3`
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`C`

Show Worked Solution

`text(By elimination:)`

`text(A single negative root occurs when)\ \ x =–1`

`->\ text(Eliminate A and D)`

`text(When)\ \ x = 0, \ y > 0`

`->\ text(Eliminate B)`

`=> C`

Functions, 2ADV F1 2014 HSC 6 MC

Which expression is a factorisation of  `8x^3 + 27`? 

  1. `(2x - 3)(4x^2 + 12x - 9)`
  2. `(2x + 3)(4x^2 - 12x + 9)`
  3. `(2x - 3)(4x^2 + 6x - 9)`
  4. `(2x + 3)(4x^2 - 6x + 9)`
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`D`

Show Worked Solution

`8x^3 + 27`

COMMENT: Factorising a cubic is only examinable with scaffolding, as provided here by expanding the answer options.

`= (2x)^3 + 3^3`

`= (2x + 3)(4x^2 – 6x + 9)`

 
`=>  D`