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GO Classes CS Test Series | Mock GATE | Test 4 | Question: 1

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Which of the following functions $f$ defined for all numbers $x$ has the property that $f(-x)=-f(x)$ for all numbers $x?$
  1. $f(x)=\frac{x^3}{x^2+1}$
  2. $f(x)=\frac{x^2-1}{x^2+1}$
  3. $f(x)=x^2\left(x^2-1\right)$
  4. $f(x)=x\left(x^3-1\right)$

1 Answer

5 5 votes
To determine which of the functions among the four choices has the property that $f(-x)=-f(x)$ for all numbers $x$, you need to check each choice until you find one that has the property.
For Option A:
$$
f(-x)=\frac{(-x)^3}{(-x)^2+1}=\frac{-x^3}{x^2+1} \text { and }-f(x)=-\left(\frac{x^3}{x^2+1}\right)=\frac{-x^3}{x^2+1} .
$$
Therefore Choice A has the property $f(-x)=-f(x)$.
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