Question

True or False The function \(f(x)=x^{-3}\) is an odd function. Justify your answer.

Short answer

True, the function \(f(x)=x^{-3}\) is an odd function because it satisfies the property \(f(-x) = -f(x)\)

Step-by-step solution

Write down the function

The given function is \(f(x)=x^{-3}\). It's necessary to verify if this function is odd.

Replace x with -x in the function

Replace every instance of x in the function with -x. This gives the new function as \(f(-x)=(-x)^{-3}\). This simplifies to \(f(-x)= -x^{-3}\).

Check if the original function and new function are negatives of each other

Now check the definition of an odd function: \(f(-x) = -f(x)\). If we negate the original function, we get \(-f(x) = -x^{-3}\), which is what we got when we replaced x with -x. This shows that our given function \(f(x)=x^{-3}\) is indeed an odd function.