Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, given each function \(f,\) evaluate \(f(-1), \quad f(0), \quad f(2),\) and \(f(4) .\) \(f(x)=\left\\{\begin{array}{ll}7 x+3 & \text { if } x<0 \\ 7 x+6 & \text { if } x \geq 0\end{array}\right.\)

Step-by-Step Solution

Answer

\(f(-1) = -4\), \(f(0) = 6\), \(f(2) = 20\), \(f(4) = 34\).

Step 1: Evaluate f(-1)

Since \(-1 < 0\), use the first part of the piecewise function: \(f(x) = 7x + 3\).Substitute \(x = -1\):\[f(-1) = 7(-1) + 3 = -7 + 3 = -4\].So, \(f(-1) = -4\).

Step 2: Evaluate f(0)

Since \(0 \geq 0\), use the second part of the piecewise function: \(f(x) = 7x + 6\).Substitute \(x = 0\):\[f(0) = 7(0) + 6 = 0 + 6 = 6\].Thus, \(f(0) = 6\).

Step 3: Evaluate f(2)

Since \(2 \geq 0\), again use the second part of the piecewise function: \(f(x) = 7x + 6\).Substitute \(x = 2\):\[f(2) = 7(2) + 6 = 14 + 6 = 20\].Hence, \(f(2) = 20\).

Step 4: Evaluate f(4)

Since \(4 \geq 0\), continue using the second part of the function: \(f(x) = 7x + 6\).Substitute \(x = 4\):\[f(4) = 7(4) + 6 = 28 + 6 = 34\].Therefore, \(f(4) = 34\).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the value of a function for given input values. In the context of piecewise functions, such as the function in the original exercise, it involves selecting the correct piece (or rule) of the function based on the input value. This is done by checking which condition the input value satisfies.

The function given, \[f(x) = \begin{cases} 7x + 3, & \text{if } x < 0 \ 7x + 6, & \text{if } x \geq 0 \end{cases} \]is a piecewise function divided into two rules. To evaluate this function at specific values (e.g., \(-1, 0, 2, 4\)), follow these steps:

• First, determine which part of the piecewise function corresponds to the input value.
• Next, substitute the input value into the selected rule to calculate the result.

For example, for \(-1\), use \(7x + 3\) because \(-1 < 0\), and calculating gives \(f(-1) = -4\). This clear breakdown of steps helps in avoiding mistakes and understanding how different parts of the function apply in different scenarios.

Algebra

Algebra is an essential branch of mathematics that facilitates working with symbols and letters to solve equations and represent numbers. When evaluating a piecewise function using algebra, the key is applying basic operations and following mathematical rules to simplify expressions.

For the function evaluation:

• Identify the correct algebraic expression from the piecewise function rules (e.g., \(7x + 3\) or \(7x + 6\)).
• Substitute the given value of \(x\) into this expression to find the result.
• Perform the arithmetic calculations step by step, ensuring accuracy in addition and multiplication.

These steps are crucial because the final result depends on proper substitution and calculation. For example, substituting \(x = 0\) in the expression \(7x + 6\) requires performing the operations: \(7(0) + 6 = 6\). Algebra simplifies the process of dealing with such functions and highlights the importance of following operations systematically.

College Mathematics

College mathematics often introduces concepts like piecewise functions, which consolidate various mathematical skills, including understanding functions, algebra, and conditional statements. It entails a deeper level of comprehension, as students must synthesize these ideas to accurately evaluate and interpret the behavior of mathematical expressions.

In this exercise:

• Students are expected to navigate different conditions within a piecewise function ( \(x < 0\) and \(x \geq 0\)).
• This requires logical reasoning to decide which part of the function to apply, an essential skill developed in higher education math courses.

Moreover, piecewise functions serve as a foundation for understanding more complex mathematical concepts, such as limits and continuity in calculus. These exercises hone students' abilities to think critically and apply mathematical theory to practical problems. Through consistent practice, students refine their problem-solving skills and prepare for advanced topics they'll encounter during their college studies.