Problem 62 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x^{3}-x$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{3} - x\) is an odd function.

Step 1: Identify the function

First, given that the function that you want to analyze is \(f(x) = x^{3} - x\). The task is to determine if this function is even, odd, or neither.

Step 2: Test for evenness

Start by testing whether it's an even function. Substitute \(-x\) for \(x\) in the function to see if the function is equivalent to the original one. \[ f(-x) = (-x)^{3} - (-x) = -x^{3} + x\] We can see that \(f(-x)\) is not equivalent to \(f(x)\), hence, the function is not even.

Step 3: Test for oddness

Now, test for oddness. The function will be odd if \(-f(x) = f(-x)\). By multiplying original function \(f(x)\) by -1, we get \[ -f(x) = -x^{3} + x \] Comparing this with our result from Step 2, we can see that \(-f(x) = f(-x)\). Therefore, the function is odd.

Step 4: Conclusion

After checking for evenness and oddness, we found that the function \(f(x) = x^{3} - x\) is odd because it satisfies the property of odd functions, namely, \(-f(x) = f(-x)\).