Problem 38 - Precalculus Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. \(f(x)=\frac{|x+3|}{x+3}\) a. \(f(5)\) b. \(f(-5)\) c. \(f(-9-x)\)

Step-by-Step Solution

Answer

The function values are \(f(5) = 1\), \(f(-5) = -1\) and \(f(-9-x)\) can be either 1 or -1, depending on the value of x.

Step 1: Evaluate \(f(5)\)

First, replace x with 5 in the function: \(f(5) = \frac{|5+3|}{5+3} = \frac{8}{8} = 1.\)

Step 2: Evaluate \(f(-5)\)

Replace x with -5: \(f(-5) = \frac{|-5+3|}{-5+3} = \frac{2}{-2} = -1.\)

Step 3: Evaluate \(f(-9-x)\)

Replace x with -9-x: \(f(-9-x) = \frac{|-9-x+3|}{-9-x+3}\). This simplifies to \(f(-9-x) = \frac{|-6-x|}{-6-x}\). Now this can either be 1 or -1, depending on the value of x. If \(-6-x\) is positive or zero, function evaluates to 1; If \(-6-x\) is negative, the function evaluates to -1.