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Chapter 2: Problem 12

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

Short Answer

Expert verified
The completed table is: \([-3, 3], [-2, 1], [0, 3], [1, 5], [3, 9]\).

Step by step solution

01

Understand the Function

The function given is a piecewise function involving an absolute value: \(g(x) = |2x + 3|\). This means we need to evaluate the expression inside the absolute value and if it is negative, take its positive version.
02

Evaluate g(-3)

Substituting \(x = -3\) into the function: \(g(-3) = |2(-3) + 3| = |-6 + 3| = |-3| = 3\). Thus, \(g(-3) = 3\).
03

Evaluate g(-2)

Substituting \(x = -2\) into the function: \(g(-2) = |2(-2) + 3| = |-4 + 3| = |-1| = 1\). Thus, \(g(-2) = 1\).
04

Evaluate g(0)

Substituting \(x = 0\) into the function: \(g(0) = |2(0) + 3| = |0 + 3| = |3| = 3\). Thus, \(g(0) = 3\).
05

Evaluate g(1)

Substituting \(x = 1\) into the function: \(g(1) = |2(1) + 3| = |2 + 3| = |5| = 5\). Thus, \(g(1) = 5\).
06

Evaluate g(3)

Substituting \(x = 3\) into the function: \(g(3) = |2(3) + 3| = |6 + 3| = |9| = 9\). Thus, \(g(3) = 9\).
07

Complete the Table

Based on our calculations, we can fill in the table:\[\begin{array}{|r|r|}\hlinex & g(x) \\hline-3 & 3 \-2 & 1 \0 & 3 \1 & 5 \3 & 9 \\hline\end{array}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Piecewise Functions
A piecewise function is a function defined by multiple sub-functions, each with its own specific domain. This means that different parts of the function apply to different intervals of the input values. In the case of the absolute value function, \(g(x) = |2x + 3|\), it is essentially split into two pieces:
  • When \(2x + 3 \geq 0\), the function behaves as \(g(x) = 2x + 3\).
  • When \(2x + 3 < 0\), we take the negative of the expression to ensure a positive outcome: \(g(x) = -(2x + 3)\).
By handling each case separately, we can understand how the function plots on different sections of the x-axis, allowing for more versatile expression than a single formula could accommodate.
Function Evaluation
Evaluating a function means calculating its value for specific input values. Let's consider the function \(g(x) = |2x + 3|\). To evaluate this function:
  • Substitute the input value into the function. For example, if \(x = -2\), plug it in to get \(g(-2) = |2(-2) + 3|\).
  • Perform the arithmetic inside the absolute value bars: \(2(-2) + 3 = -4 + 3 = -1\).
  • Since the absolute value makes all numbers positive, convert the result to \(1\). Hence, \(g(-2) = 1\).
This step is repeated for every input value given in the problem. The process involves basic algebra followed by applying the absolute value operation.
Input-Output Table
An input-output table is a simple way to organize values of a function for different inputs. In this exercise involving the function \(g(x) = |2x + 3|\), various x-values were used to calculate their corresponding outputs or g(x) values.
  • The left column signifies the input values (\(x\)) provided in the problem, like -3, -2, 0, 1, and 3.
  • The right column is filled by computing the function value for each input. This is the g(x) value, which in our completed table are 3, 1, 3, 5, and 9 respectively.
This approach helps visualize and verify each function evaluation systematically, reinforcing the understanding of how different inputs affect the output.

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