Question

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. $$ \begin{array}{l} f(x)=x^{3}-x \\ \frac{f(x)-f(1)}{x-1} \end{array} $$

Short answer

The simplified version of the given expression \(\frac{f(x) - f(1)}{x - 1}\) is \(x(x + 1)\).

Step-by-step solution

Find \(f(1)\)

This means substituting \(x\) with \(1\) in the function \(f(x)\). So, \(f(1) = (1)^{3} - 1 = 1 - 1 = 0

Substitute \(f(x)\) and \(f(1)\) into the expression

Substitute \(f(x)= x^{3} - x\) and \(f(1)=0\) into \(\frac{f(x) - f(1)}{x-1}\). The result is \(\frac{x^{3} - x - 0}{x-1}\) which simplifies to \(\frac{x^{3} - x}{x - 1}\)

Simplify the expression

To simplify \(\frac{x^{3} - x}{x - 1}\), factor out \(x\) from the numerator. The resulting expression is \(\frac{x(x^{2} - 1)}{x - 1}\). The expression \(x^{2} - 1\) can be factored further, this is a difference of squares, into \(x-1\) and \(x+1\). This reduces the full expression to \(\frac{x[(x-1)(x+1)]}{x - 1}\). By cancelling out \(x-1\) from the numerator and denominator, we're left with the simplified expression \(x(x + 1)\)