smc-987-70-Odd/even functions

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

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

\(\therefore f(x)\ \text{is odd.}\)

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

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

\(\therefore f(x)\ \text{is odd.}\)

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

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

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\(D\)

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

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

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

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

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

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

\(=>D\)

♦♦♦ Mean mark 24%.

Which diagram shows the graph of an odd function?

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`A`

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`text(Odd functions occur when:)`

♦ Mean mark 38%.

`f(x) = – f(x)`

`text(Graphically, this occurs when a function has)`

`text(symmetry when rotated 180° about the origin.)`

`=> A`