Problem 21 - Contemporary Precalculus

Question

Each equation defines y as a function of \(x .\) Create a table that shows the values of the function for the given values of \(x\) $$y=\left|x^{2}-5\right| ; \quad x=-8,-6, \ldots, 8,10,12$$

Step-by-Step Solution

Answer

Short answer: For the function \(y = \left|x^{2}-5\right|\), we calculated the values of \(y\) for each specified value of \(x\) and put them in a table. The table is as follows: Value of x | Value of y -----------|----------- -12 | 149 -10 | 105 -8 | 69 -6 | 41 -4 | 21 -2 | 9 0 | 5 2 | 1 4 | 11 6 | 31 8 | 59 10 | 95 12 | 139

Step 1: Set up the table

To begin, create a table with two rows: one for the values of \(x\), and one for the corresponding values of \(y\), as defined by the given function.

Step 2: Calculate the values of y for each x value

Calculate the value of \(y\) for each value of \(x\) using the given function \(y = \left|x^{2}-5\right|\).

Step 3: Fill in the table with the computed values

Complete the table as you calculate the function values: Value of x | Value of y -----------|----------- -12 | y = \$@\left|-12^2-5\right| = |-144-5| = 149@$@ -10 | y = \$@\left|-10^2-5\right| = |-100-5| = 105@$@ -8 | y = \$@\left|-8^2-5\right| = |-64-5| = 69@$@ -6 | y = \$@\left|-6^2-5\right| = |-36-5| = 41@$@ -4 | y = \$@\left|-4^2-5\right| = |-16-5| = 21@$@ -2 | y = \$@\left|-2^2-5\right| = |-4-5| = 9@$@ 0 | y = \$@\left|0^2-5\right| = |-5| = 5@$@ 2 | y = \$@\left|2^2-5\right| = |-1| = 1@$@ 4 | y = \$@\left|4^2-5\right| = |11| = 11@$@ 6 | y = \$@\left|6^2-5\right| = |31| = 31@$@ 8 | y = \$@\left|8^2-5\right| = |59| = 59@$@ 10 | y = \$@\left|10^2-5\right| = |95| = 95@$@ 12 | y = \$@\left|12^2-5\right| = |139| = 139@$@ Now we have a complete table showing the function values for each specified \(x\) value.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for specific input values. In this exercise, we need to evaluate the function defined by the expression:\[ y = \left|x^2 - 5\right| \]where \(x\) is a variable, and \(y\) is the function's output. Evaluating a function involves substituting different values of \(x\) into the expression and simplifying to find the corresponding \(y\) values.
This process helps us understand how changes in \(x\) affect \(y\). It is crucial for graphing functions and solving real-world problems.

Creating a Table

Creating a table is a helpful way to organize and display function evaluations. It simplifies the task of calculating and comparing values. Here, a table is used to clearly state the function \(y = \left|x^2 - 5\right|\) for various values of \(x\).
The table has two columns:

• One for the input values of \(x\)
• One for the output values \(y\)

By listing the values this way, we can easily observe the pattern or relationship between \(x\) and \(y\). This organized visual representation makes it easier to check calculations or utilize the data for further analysis.

Mathematical Expressions

Mathematical expressions, like \(y = \left|x^2 - 5\right|\), describe the relationships between variables using mathematical symbols and operations. The expression in this exercise uses absolute value to ensure the result is always non-negative, no matter the input \(x\).
Breaking down the expression:

• \(x^2\) squares the input \(x\), which affects how steeply \(y\) changes as \(x\) increases or decreases.
• Subtracting 5 shifts the function downward on a graph.
• The absolute value \(\left| ... \right|\) ensures that no matter if \(x^2 - 5\) results in a positive or a negative, \(y\) will always be positive or zero.

Understanding these expressions is crucial for grasping how different elements of a function affect its behavior.