Question

If \(f(x)=2 x^{3}-x\), find \(f(-1), f(0), f\left(x^{2}\right), f(\sqrt{x})\) and \(f\left(\frac{1}{x}\right)\).

Short answer

The short answers are: \(f(-1) = -1\), \(f(0) = 0\), \(f(x^2) = 2x^6 - x^2\), \(f(\sqrt{x}) = 2x^{3/2} - x^{1/2}\), and \(f\left(\frac{1}{x}\right) = \frac{2}{x^3} - \frac{1}{x}\).

Step-by-step solution

Evaluating f(-1)

To find \(f(-1)\), replace \(x\) with \(-1\) in the formula for \(f(x)\): \[f(-1) = 2(-1)^3 - (-1)\] Now, simplify the expression: \[f(-1) = 2(-1) + 1\] \[f(-1) = -2 + 1\] \[f(-1) = -1\]

Evaluating f(0)

To find \(f(0)\), replace \(x\) with \(0\) in the formula for \(f(x)\): \[f(0) = 2(0)^3 - (0)\] Now, simplify the expression: \[f(0) = 0\]

Evaluating \(f(x^{2})\)

To find \(f(x^2)\), replace \(x\) with \(x^2\) in the formula for \(f(x)\): \[f(x^2) = 2(x^2)^3 - (x^2)\] Now, simplify the expression: \[f(x^2) = 2x^6 - x^2\]

Evaluating \(f(\sqrt{x})\)

To find \(f(\sqrt{x})\), replace \(x\) with \(\sqrt{x}\) in the formula for \(f(x)\): \[f(\sqrt{x}) = 2(\sqrt{x})^3 - (\sqrt{x})\] Now, simplify the expression: \[f(\sqrt{x}) = 2x^{3/2} - x^{1/2}\]

Evaluating \(f(\frac{1}{x})\)

To find \(f(\frac{1}{x})\), replace \(x\) with \(\frac{1}{x}\) in the formula for \(f(x)\): \[f\left(\frac{1}{x}\right) = 2\left(\frac{1}{x}\right)^3 - \left(\frac{1}{x}\right)\] Now, simplify the expression: \[f\left(\frac{1}{x}\right) = 2\left(\frac{1}{x^3}\right) - \left(\frac{1}{x}\right)\] \[f\left(\frac{1}{x}\right) = \frac{2}{x^3} - \frac{1}{x}\] Now, we found all the requested values and expressions for \(f(x)\).