Problem 38 - College Algebra Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(f(x)=\left\\{\begin{array}{lll}6 x-1 & \text { if } & x<0 \\ 7 x+3 & \text { if } & x \geq 0\end{array}\right.\) \(f(-3)\) b. \(f(0)\) c. \(f(4)\)

Step-by-Step Solution

Answer

The evaluated functions are: a. \(f(-3) = -19\), b. \(f(0) = 3\), c. \(f(4) = 31\)

Step 1: Evaluate f(-3)

Observing that -3 is less than 0, the form \(6x - 1\) should be used to evaluate this function. Substituting -3 for x gives \(6*(-3) - 1\) which equals -19.

Step 2: Evaluate f(0)

Observing that 0 is equal to 0, the form \(7x + 3\) should be used to evaluate this function. Substituting 0 for x gives \(7*0 + 3\) which equals 3.

Step 3: Evaluate f(4)

Observing that 4 is greater than 0, the form \(7x + 3\) should be used to evaluate this function. Substituting 4 for x gives \(7*4 + 3\) which equals 31.

Key Concepts

Independent Variable

In mathematics, the independent variable is a crucial concept, especially when working with functions like piecewise functions. It acts as the input value that you feed into a function to determine the output. Think of it as a variable you can manipulate to explore how the function behaves under different circumstances.
In our example, the independent variable is denoted by \(x\). This means that for different values of \(x\), the function \(f(x)\) will potentially exhibit different behaviors, depending on the rules set by the piecewise definition. For instance, when evaluating \(f(-3)\), \(f(0)\), and \(f(4)\), each value of \(x\) points to a different segment of the function's definition based on its conditions, like whether \(x\) is less than zero, equal to zero, or greater than zero.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. In piecewise functions, this involves determining which segment of the function's definition applies based on the value of the independent variable.

• For \(x = -3\), since \(-3 < 0\), we use the expression \(6x - 1\) to evaluate the function. Substituting \(-3\) yields \(6(-3) - 1 = -19\).
• For \(x = 0\), since \(0 \geq 0\), we use \(7x + 3\). Substituting \(0\) results in \(7(0) + 3 = 3\).
• For \(x = 4\), as \(4 \geq 0\), we again use \(7x + 3\). Substituting \(4\) gives \(7(4) + 3 = 31\).

Each evaluation follows the particular piece of the function tailored to the independent variable's condition, highlighting the importance of correctly choosing the function's section.

Algebraic Expressions

Algebraic expressions form the backbone of piecewise functions. They are the mathematical phrases that help in creating the rules inside each piece of the piecewise function. Each piece consists of an algebraic expression tied to a condition for the independent variable.
In the given example, the piecewise function employs two different algebraic expressions:

• \(6x - 1\) used when \(x < 0\)
• \(7x + 3\) used when \(x \geq 0\)

These expressions dictate the output of the function based on defined rules. The operations within these expressions involve core algebraic techniques like multiplication and addition, which are essential for evaluating functions. Understanding how to manipulate and substitute values into these algebraic forms is vital for accurately assessing the piecewise function's behaviors across its different segments.