Problem 39 - College Algebra Essentials

Question

Evaluate each piece wise 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. \(g(-6) \quad\) c. \(g(-3)\)

Step-by-Step Solution

Answer

The values of the piecewise function at the given points are a. \(g(0) = 3\), b. \(g(-6) = 3\), c. \(g(-3) = 0\).

Step 1: Evaluate g(0)

First, check which condition the input value applies to. Here, \(0 \geq -3\), so the sub-function corresponding to this condition is used: \(g(0) = 0 + 3 = 3\).

Step 2: Evaluate g(-6)

In this case, \(-6 < -3\), so the sub-function corresponding to this condition is used. Calculate the value: \(g(-6) = -((-6) + 3) = -(-3) = 3\).

Step 3: Evaluate g(-3)

For \(x = -3\), use the first sub-function because the condition states \(x \geq -3\). So, \(g(-3) = -3 + 3 = 0\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions constructed using a finite number of algebraic operations, such as addition, subtraction, multiplication, division, and roots, among others. These operations are carried out on variables and numbers to define a particular relationship.

For instance, in the piecewise function featured in the exercise, we have two different algebraic expressions defined based on the value of the independent variable, which is typically denoted as 'x'. The function changes its algebraic formula depending on the condition that the input value of 'x' meets. Understanding algebraic functions is crucial because they are the building blocks for describing patterns, relationships, and changes within various scientific and mathematical contexts.

The function presented in the exercise can be described as follows:

• If the input 'x' is greater than or equal to -3, the output or function value is defined by 'x+3'.
• If 'x' is less than -3, the output is defined by '-(x+3)'.

Function Evaluation

Function evaluation entails finding the value of a function for a specific input. It is akin to substituting a given number into an equation to calculate the result. Evaluating a function tells you the output that corresponds to a particular input.

In the context of the exercise, evaluating the piecewise function, ‘g(x)’, involves determining which algebraic expression to use for the given values of 'x' (0, -6, and -3). By doing so, you calculate the output of the function for these inputs. For piecewise functions, this process requires an extra step – before substituting the value into the appropriate function expression, the evaluator must first decide which piece of the function applies.

This requires:

• Checking the conditions associated with each piece of the function.
• Identifying which condition the input value satisfies.
• Applying the input to the respective algebraic expression of the function.

Conditional Statements in Functions

Conditional statements in functions are rules that define different outputs or expressions of a function based on certain conditions or ranges of the independent variable. They are crucial in piecewise functions as they dictate which part of the function to use to evaluate an input value.

Conditional statements are usually presented in a 'if-then' format. For the given piecewise function, 'g(x)', we encounter two conditions:

• The 'if' part for the first condition is 'x ≥ -3', which then defines the function to be 'x + 3'.
• For the second condition, the 'if' part is 'x < -3', which changes the function to '-(x + 3)'.

Understanding how to apply these conditions is vital to function evaluation and occurs in many mathematical scenarios beyond piecewise functions as well, such as when determining whether a value falls within a certain range for real-world applications and modeling.

Therefore, it is essential for students to grasp the way conditional statements dictate the form and output of piecewise functions, as well as other types of functions that incorporate conditional logic.