Problem 40 - College Algebra Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(g(x)=\left\\{\begin{array}{ll}x+5 & \text { if } x \geq-5 \\ -(x+5) & \text { if } x<-5\end{array}\right.\) a. \(g(0)\) b. \(g(-6)\) c. \(g(-5)\)

Step-by-Step Solution

Answer

The values of \(g(0)\), \(g(-6)\), and \(g(-5)\) are 5, 1, and 0, respectively.

Step 1: Evaluate g(0)

First identify which rule to use given the value of x (0 in this case). Since \(0 \geq -5\), the rule \(g(x) = x + 5\) applies. Substitute 0 into this equation to get \(g(0) = 0 + 5 = 5\).

Step 2: Evaluate g(-6)

Next, proceed with the second value of x, which is -6. This time, since \(-6 < -5\), the rule \(g(x) = -(x + 5)\) applies. Substitute -6 into this equation to get \(g(-6) = -(-6 + 5) = -(-1) = 1\).

Step 3: Evaluate g(-5)

Lastly, for \(x = -5\), since \(-5 \geq -5\), the rule \(g(x) = x + 5\) again applies. Substitute -5 into this equation to get \(g(-5) = -5 + 5 = 0\).

Key Concepts

Evaluating Functions

Understanding how to evaluate functions is a fundamental skill in algebra. When students come across evaluating functions, they are being asked to determine the outcome of a function given a specific input. In this context, the function acts like a machine: for every input (an independent variable or simply an 'x' value), there is a corresponding output (a dependent variable or a 'y' value).

Following the exercise example, when evaluating the function for a given value, such as finding the value of g(0), the process starts with identifying which part of the piecewise function applies. This step is crucial because the function's definition changes depending on what input value you are using. This means you should substitute the input value into the correct rule—a rule that dictates the function’s behavior for a specific interval of 'x' values—and solve for the function's output.

Piecewise Function Notation

Piecewise functions are a type of function that have different expressions based on the input value. The piecewise function notation breaks down a function into two or more 'pieces' that each have their own rules. Each 'piece' of the function is defined by a condition that the independent variable must meet.

For instance, in our given exercise, the function g(x) is defined by two rules: one rule for when x is greater than or equal to -5, and another for when x is less than -5. Visually, this is represented by a curly brace that groups the separate rules together. To evaluate piecewise functions correctly, one must pay attention to these conditions and apply the appropriate rule for the given value of x.

Function Value Calculation

The function value calculation revolves around substituting the specified input value of the variable into the appropriate equation, and simplifying it to obtain the output value. In our exercise, calculating the value of g(-6) required using the second rule -(x + 5) because the input, -6, satisfied the condition for that rule (it was less than -5). After substituting -6 into the equation, the simplification process led to a function value of 1.

This step-by-step process emphasizes the importance of firstly understanding which condition the input satisfies and secondly performing the substitution and simplification carefully to arrive at the correct output. By mastering these techniques, one can confidently tackle more complex functions and further their algebraic skills.