Problem 97 - Precalculus Essentials

Question

Find \(f(-x)-f(x)\) for the given function \(f\) Then simplify the expression. $$f(x)=x^{3}+x-5$$

Step-by-Step Solution

Answer

The simplified expression for \(f(-x)-f(x)\) is \(-2x^{3} - 2x\)

Step 1: Substitute -x into the function

The function given is \(f(x)=x^{3}+x-5\). We substitute -x in place of x to get \(f(-x)=(-x)^{3}+(-x)-5=-x^{3}-x-5\)

Step 2: Substitute x into the function

Substitute the x into the function to get \(f(x)=x^{3}+x-5\)

Step 3: Find the difference

We're asked to find \(f(-x)-f(x)\), so we subtract \(x^{3}+x-5\) from \(-x^{3}-x-5\) to get \(-x^{3}-x-5 - (x^{3}+x-5)\)

Step 4: Simplify the expression

Simplify the expression obtained in the previous step by removing parentheses and collecting like terms to get \(-x^{3} - x - 5 - x^{3} - x + 5\) which simplifies to \(-2x^{3} - 2x\)