Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
Determine whether a function is odd by comparing \(f(-x)\) and \(-f(x)\)
Problem
Determine whether \(f(x)=x^{3}\) is odd. Compute \(f(-x)\) and \(-f(x)\) separately, compare the simplified expressions using \(f(-x)=-f(x)\), and connect the conclusion to origin symmetry.
Big Picture
What this problem is really about
Oddness compares the opposite-input output with the negative of the original output, and both expressions should be computed separately. We’ll simplify f of negative x and negative f of x, test their identity across the domain, and connect the result to origin symmetry.
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