Problem 12 - Algebra and Trigonometry

Question

Complete the table. $$ g(x)=|2 x+3| $$ $$ \begin{array}{|c|c|}\hline x & {g(x)} \\ \hline-3 & {} \\ {-2} & {} \\ {0} & {} \\ {1} & {} \\ {3} & {} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The completed table has values: \((-3, 3), (-2, 1), (0, 3), (1, 5), (3, 9)\).

Step 1: Understand the Function

The function given is \( g(x) = |2x + 3| \). The absolute value function outputs the non-negative value of the expression inside the absolute value bars, \( |a| = a \) if \( a \geq 0 \) and \( |a| = -a \) if \( a < 0 \).

Step 2: Evaluate for \( x = -3 \)

Substitute \( x = -3 \) into the function: \( g(-3) = |2(-3) + 3| = |-6 + 3| = |-3| = 3 \).

Step 3: Evaluate for \( x = -2 \)

Substitute \( x = -2 \) into the function: \( g(-2) = |2(-2) + 3| = |-4 + 3| = |-1| = 1 \).

Step 4: Evaluate for \( x = 0 \)

Substitute \( x = 0 \) into the function: \( g(0) = |2(0) + 3| = |0 + 3| = |3| = 3 \).

Step 5: Evaluate for \( x = 1 \)

Substitute \( x = 1 \) into the function: \( g(1) = |2(1) + 3| = |2 + 3| = |5| = 5 \).

Step 6: Evaluate for \( x = 3 \)

Substitute \( x = 3 \) into the function: \( g(3) = |2(3) + 3| = |6 + 3| = |9| = 9 \).

Step 7: Complete the Table

Now that we have evaluated \( g(x) \) for each \( x \) value, complete the table:\[ \begin{array}{|c|c|}\hline x & g(x) \\hline -3 & 3 \-2 & 1 \0 & 3 \1 & 5 \3 & 9 \\hline \end{array} \]

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different formulas or rules over different intervals or domains of the input variable. For the function to be a piecewise function, it must have different expressions for different intervals. This type of function is useful to represent situations where a rule or formula changes based on the input value you choose. A common example of a piecewise function is the absolute value function.The absolute value function, such as in the exercise, can be thought of as a piecewise function. It follows different rules based on whether the input is non-negative or negative:

• If the input is non-negative, the absolute value function returns the input as is.
• If the input is negative, it returns the negative of the input, effectively making it positive.

This makes the function look like this as a piecewise:\[g(x) = \begin{cases}2x + 3, & \text{if } 2x + 3 \geq 0\-(2x + 3), & \text{if } 2x + 3 < 0\end{cases}\]By understanding this, the function is easier to evaluate piece by piece.

Function Evaluation

Function evaluation means finding the output of a function for specific input values. It's like using a function as a machine: you input numbers, and it processes them according to a set rule, then gives you an output. In algebra, function evaluation is a fundamental skill.To evaluate the function given in the exercise, you substitute each value of \( x \) into the function \( g(x) = |2x + 3| \) and simplify.

Steps to Evaluate a Function

Here's how you evaluate \( g(x) \) for different \( x \) values:

• Substitute the \( x \) value into the expression.
• Perform the arithmetic inside the absolute value.
• Apply the absolute value by determining if the result is positive or negative.
• Output the absolute value result.

For instance, when evaluating \( g(-3) \), you replace \( x \) with -3: 1. Inside: \( 2(-3) + 3 = -6 + 3 = -3 \)2. Absolute value: \( |-3| = 3 \)These steps can be applied to every value in the table to determine each output.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, or division) without an equal sign. They're like parts of a puzzle in algebra. You use them to build equations and functions. In the context of the exercise, we are dealing with the expression \( 2x + 3 \) within the absolute value function.

Simplifying Algebraic Expressions

Simplifying these expressions often involves computing with numbers and variables step by step:

• Apply the operations in the correct order (remember the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
• Combine like terms if possible, though in this exercise it was more about replacing the \( x \) values with specific numbers.

For the function \( g(x) = |2x + 3| \), the algebraic expression \( 2x + 3 \) is key before applying the absolute value. Substituting different values into this expression helps us to understand its role more explicitly—by showing how it changes across different \( x \) values before being modified by the absolute value rule.