Problem 39 - College Algebra

Question

Evaluate each piecewise function at the given values of the independent variable. $$g(x)=\left\\{\begin{array}{ll}x+3 & \text { if } x \geq-3 \\\\-(x+3) & \text { if } x<-3\end{array}\right.$$ $$a. g(0)$$ $$b. g(-6)$$ $$c. g(-3)$$

Step-by-Step Solution

Answer

The evaluated values are: a) \(g(0) = 3\), b) \(g(-6) = 3\), and c) \(g(-3) = 0\).

Step 1: Evaluate g(0)

The value of \(x\) is 0. Looking at the conditions for our piecewise function, we see that 0 is greater than or equal to -3. So, we use the function \(x + 3\). Substituting \(x\) with 0, we have \(g(0) = 0 + 3 = 3\).

Step 2: Evaluate g(-6)

The value of \(x\) is -6. Our conditions state that if \(x\) is less than -3, then we use the function \(- (x + 3)\). So, \(g(-6) = - (-6 + 3) = - (-3) = 3\).

Step 3: Evaluate g(-3)

The value of \(x\) is -3. We see that -3 is equal to -3, so we use the function \(x + 3\). Thus, \(g(-3) = -3 + 3 = 0\).

Key Concepts

Evaluating Functions

When we evaluate functions, we're essentially substituting the independent variable with a given value to find the output. Think of a function as a machine: you put a number in, the machine works its magic according to a set of rules, and then a new number comes out. In mathematical terms, these 'magic rules' are the operations defined by the function.
For example, if we have a function defined as \(f(x) = 2x + 1\), evaluating \(f(3)\) means replacing every \(x\) in the function with 3, resulting in \(2(3) + 1 = 7\). The piecewise function in our exercise has two different 'rules' depending on the value of \(x\). When we evaluate the piecewise function at specific points, like 0, -6, or -3, we choose the appropriate rule and plug in the value to get our answer.

Evaluation with the Step-by-Step Example:

• To evaluate \(g(0)\), since 0 is equal to or greater than -3, we use the first function condition, which gives us \(g(0) = 0 + 3 = 3\).
• For \(g(-6)\), -6 is less than -3, triggering the second condition, resulting in \(g(-6) = - (-6 + 3) = 3\).
• Last, \(g(-3)\) is right on the border where \(x\) is -3, so we use the first condition because it includes values equal to -3, which simplifies to \(g(-3) = -3 + 3 = 0\).

Understanding how to evaluate functions is crucial, as it's a foundational skill in algebra that allows us to work with any kind of functions, not just piecewise ones.

Independent Variable

The independent variable is the input of a function, usually represented by \(x\). It's called 'independent' because it can vary freely; we can choose its value without considering the function's rules. The function's output, often represented by \(y\) or \(f(x)\), is called the dependent variable because its value depends on the chosen input.
In a piecewise function, the independent variable still goes into the function, but the output not only depends on the rules corresponding to the variable's value but also on which rule applies to the input range. In our exercise, \(g(x)\), the independent variable \(x\) is evaluated under different conditions depending on whether \(x\) is less than or greater than/equal to -3.

Why Knowing the Independent Variable Matters:

• Identifies which part of the piecewise function to use.
• Helps in graphing functions by determining the appropriate section of the graph.
• Is essential in understanding function behavior and calculating outputs.

Understanding the role of the independent variable enables students to solve problems involving not just piecewise functions, but any mathematical functions they encounter.

Function Conditions

Function conditions are the rules that dictate which part of a piecewise function to use in relation to the independent variable's value. Essentially, they're the 'instructions' for what to do with your input (independent variable) under different circumstances.
Each piece in a piecewise function has its own condition. These conditions can be inequalities or specific values that tell us when to apply a particular rule. It's like having multiple functions in one, with clear boundaries for when each function comes into play.

Understanding Function Conditions in Our Exercise:

• If \(x \geq -3\), we use the condition \(x + 3\); this part of the function deals with inputs that are greater than or equal to -3.
• If \(x < -3\), the condition is \(-(x + 3)\), so for any input less than -3, we flip the sign and add 3 to our negative input.

Recognizing and applying the correct function condition is vital. It ensures that each input is processed correctly and that the corresponding outputs are accurate. It's especially important for piecewise functions, which can have different behaviors depending on the input value. By mastering function conditions, students can effectively manage complex functions and apply them properly whether they're evaluating expressions or graphing functions.