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Suppose that $f(x)$ is a non-zero odd function - HSC - SSCE Mathematics Extension 2 - Question 8 - 2017 - Paper 1

Question 8

Suppose that $f(x)$ is a non-zero odd function. Which of the functions below is also odd? A. $f(x^2) ext{cos } x$ B. $f(f(x))$ C. $f(x^3) ext{sin } x$ D. $f(x^2) -... show full transcript

Worked Solution & Example Answer:Suppose that $f(x)$ is a non-zero odd function - HSC - SSCE Mathematics Extension 2 - Question 8 - 2017 - Paper 1

Step 1

A. $f(x^2) \text{cos } x$

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Answer

To determine if this function is odd, we check:

$f(-x^2) \text{cos}(-x) = -f(x^2) \text{cos}(x)$

Since $f(x^2)$ is an even function (as it depends on $x^2$ ) and $ext{cos } x$ is even, the product is not odd.

Step 2

B. $f(f(x))$

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Answer

Since $f(x)$ is an odd function, we check:

$f(f(-x)) = f(-f(x)) = -f(f(x))$

Thus, $f(f(x))$ is also odd.

Step 3

C. $f(x^3) \text{sin } x$

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Answer

Checking its oddness:

$f(-x^3) \text{sin}(-x) = -f(x^3)(- ext{sin}(x)) = f(x^3) \text{sin} x$

Here, $f(x^3)$ is odd and $ext{sin } x$ is odd, therefore the product is odd.

Step 4

D. $f(x^2) - f(x)$

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Answer

Checking:

$f(-(x^2)) - f(-x) = f(x^2) + f(x)$

Since $f(x^2)$ is even and $f(x)$ is odd, this does not yield an odd function.

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