Problem 47 - 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 values for the function evaluated at the given points are:\na. \(g(0) = 3\)\nb. \(g(-6) = 3\)\nc. \(g(-3) = 0\)

Step 1: Evaluate \(g(0)\)

Since \(0 \geq -3\), the piecewise function is defined as \(x + 3\). Thus, for \(g(0)\), replace \(x\) with \(0\) in the equation. Hence, \(g(0) = 0 + 3 = 3\).

Step 2: Evaluate \(g(-6)\)

Because \(-6 < -3\), use the function definition \(- (x+3)\) to calculate \(g(-6)\). Hence, replace \(x\) with \(-6\) to obtain \(g(-6) = - (-6 + 3) = - (-3) = 3\).

Step 3: Evaluate \(g(-3)\)

At \(x = -3\), the piecewise function could use either of the function parts, as it's defined for both \(-3 == -3\) and \(-3 \geq -3\). However, piecewise functions are usually exclusive, meaning they do not overlap. So, the common practice is to choose the equation for \(x \geq -3\). Here, \(g(-3) = -3 + 3 = 0\).

Key Concepts

Evaluate Piecewise Functions

A piecewise function is a function that is defined by different expressions depending on the input or the independent variable.
The way to evaluate piecewise functions is by carefully choosing which part of the function to use, based on the given conditions for the independent variable. To evaluate a piecewise function:

• Check the value of the independent variable.
• Determine which condition or "piece" the variable fulfills.
• Use the corresponding expression to calculate the result.

Let's consider the function from the exercise, \[g(x) = \begin{cases} x+3 & \text{if} \ x \geq -3 \ -(x+3) & \text{if} \ x < -3 \end{cases}\]Suppose you want to evaluate this function at \(x = 0\). Since \(0 \geq -3\), we pick the expression \(x + 3\). Calculating \(0 + 3\), we find \(g(0) = 3\).
This is how we effectively choose and compute a piecewise function.

Independent Variable in Functions

The independent variable in a function is the input value upon which the output or dependent variable is based. It’s often denoted as \(x\) in functions like \(f(x)\) or \(g(x)\). In piecewise functions, the independent variable plays a crucial role in determining which equation gets used to calculate the output. For the function provided:

• \(g(x)\): \(x\) is the independent variable.
• The value of \(x\) will determine whether to use the expression \(x + 3\) or \(- (x + 3)\).

You must know the boundaries and conditions tied to the independent variable to correctly evaluate the function for given inputs. If the independent variable doesn't clearly satisfy a condition, refer to how the piecewise function is designed to address overlapping conditions.

Function Notation

Function notation is the way functions are expressed in mathematical terms and equations. It provides a clear framework for understanding and evaluating functions. Usually, a function is written as \(f(x)\), where \(f\) represents the function name, and \(x\) is the independent variable.In our example, the function is denoted as \(g(x)\). Here's how function notation helps:

• The notation gives clear guidance on how to substitute the variable with actual values, for example, \(g(0)\) or \(g(-6)\).
• Function notation clarifies that the same process or rule applies universally to all inputs that meet the specified conditions.

This makes it simpler to understand and solve problems by clearly indicating which variable to evaluate and which formulas to use based on those values.

Piecewise Function Conditions

Piecewise function conditions are the specific criteria that determine which piece of the function to use. These conditions are typically based on the value of the independent variable. They set the rules or guidelines to ensure that each segment of the function is used correctly.When dealing with a piecewise function like:\[g(x) = \begin{cases} x+3 & \text{if} \ x \geq -3 \ -(x+3) & \text{if} \ x < -3 \end{cases}\]These conditions tell you:

• If \(x\) is greater than or equal to \(-3\), use \(x + 3\).
• If \(x\) is less than \(-3\), use \(- (x + 3)\).

The conditions guide you to decide which part of the function should be applied when computing outputs like \(g(0)\), \(g(-6)\), or \(g(-3)\).
By following these rules, you ensure mathematical accuracy and logical consistency in evaluating functions.