Problem 36 - Precalculus Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. \(f(x)=\frac{4 x^{3}+1}{x^{3}}\) a. \(f(2)\) b. \(f(-2)\) c. \(f(-x)\)

Step-by-Step Solution

Answer

\(f(2) = \frac{33}{8}, f(-2) = \frac{31}{8}, f(-x) = \frac{4x^{3}-1}{x^{3}}\).

Step 1: Evaluate f(2)

Plug in 2 for x in the function, i.e., \(f(2)=\frac{4*(2)^{3}+1}{(2)^{3}}\). By simplifying this expression, we get \(f(2) = \frac{4*8+1}{8} = \frac{33}{8}\).

Step 2: Evaluate f(-2)

Next, plug in -2 for x in the function, i.e., \(f(-2)=\frac{4*(-2)^{3}+1}{(-2)^{3}}\). By simplifying this expression, we get \(f(-2) = \frac{4*(-8)+1}{-8} = \frac{-32+1}{-8} = \frac{-31}{-8} = \frac{31}{8}\).

Step 3: Evaluate f(-x)

Lastly, substitute -x into the function for x, that is, \(f(-x) = \frac{4*(-x)^{3}+1}{(-x)^{3}}\). Simplifying this expression gives us \(f(-x) = \frac{-4x^{3}+1}{-x^{3}}\). We can simplify further by multiplying the top and bottom by -1. This gives us \(f(-x) = \frac{4x^{3}-1}{x^{3}}\).