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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)

Show Answers Only

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

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

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

 
\(\therefore f(x)\ \text{is odd.}\)

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 9 MC

Let \(f(x)\) be any function with domain all real numbers.

Which of the following is an even function, regardless of the choice of \(f(x)\)?

  1. \(2 f(x)\)
  2. \(f(f(x))\)
  3. \((f(-x))^2\)
  4. \(f(x) f(-x)\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Even function}\ \rightarrow \ f(x)=f(-x)\)

\(\text{Consider the function}\ \ f(x) = x-2\)

\( 2f(1)=-2,\ \ 2f(-1)=-6\ \ \text{(not even)}\)

\( f(f(1))=f(-1)=-3,\ \ f(f(-1))=f(-3)=-5\ \ \text{(not even)}\)

\( (f(-1))^2=(-3)^2=9,\ \ (f(1))^2=(-1)^2=1\ \ \text{(not even)}\)

\( f(1)f(-1)=-1 \times -3=3,\ \ f(-1)f(1)=-3 \times -1=3 \ \text{(possibly even)}\)

\(=>D\)

♦♦♦ Mean mark 24%.