Problem 112 - Algebra and Trigonometry

Question

will help you prepare for the material covered in the first section of the next chapter. If \(y=|x+1|,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-4\) and ending with 2

Step-by-Step Solution

Answer

The corresponding values of \(y = |x + 1|\) for each integer from \(-4\) to \(2\) are as follows: \(3, 2, 1, 0, 1, 2, 3\)

Step 1: Understanding the problem

We're asked to find the value of the expression \(y=|x+1|\) for each integer from \(-4\) to \(2\). It's important to understand what absolute values mean to solve this task.

Step 2: Evaluating the function for each integer

Now, let's plug these values of \(x\) into the function one by one.

Step 3: Evaluating the function for \(x=-4\)

Substituting \(x=-4\) into the function gives us \(y = |-4+1| = |-3| = 3\)

Step 4: Evaluating the function for \(x=-3\)

Substituting \(x=-3\) into the function gives us \(y = |-3+1| = |-2| = 2\)

Step 5: Evaluating the function for \(x=-2\)

Substituting \(x=-2\) into the function gives us \(y = |-2+1| = |-1| = 1\)

Step 6: Evaluating the function for \(x=-1\)

Substituting \(x=-1\) into the function gives us \(y = |-1+1| = |0| = 0\)

Step 7: Evaluating the function for \(x=0\)

Substituting \(x=0\) into the function gives us \(y = |0+1| = |1| = 1\)

Step 8: Evaluating the function for \(x=1\)

Substituting \(x=1\) into the function gives us \(y = |1+1| = |2| = 2\)

Step 9: Evaluating the function for \(x=2\)

Substituting \(x=2\) into the function gives us \(y = |2+1| = |3| = 3\)

Key Concepts

Integer Evaluation

Understanding integer evaluation is crucial when working with functions and expressions, especially those involving absolute values. Let's break down this process step-by-step. We are tasked with finding the value of the function for each integer within a given range. In this case, we start from \(-4\) and end at \(2\). To evaluate integers:

• Identify the range of integers needed for the function.
• Substitute each integer one at a time into the given function expression.
• Compute the result for the function of each integer.

This ensures that all possible integer values within the specified range are considered, allowing us to analyze all outcomes of the function.

Function Substitution

Function substitution is the method of plugging in specific values for variables in a function. Here, the function is expressed as \( y = |x + 1 | \), where \( x \) is replaced by each integer value. For example, to find the value of \( y \) at \( x = -4 \), replace every \( x \) in the expression with \(-4\). This results in evaluating \( y = |-4+1| \). Here’s a simple guide to follow for substitution:

• Take the function \( y = |x+1| \).
• Replace \( x \) with each integer starting from the minimum to the maximum in the range.
• Simplify the resulting expression to evaluate \( y \).

Substituting different integer values allows us to calculate the corresponding \( y \) values for the entire range of interest.

Step-by-Step Solution

Working through a step-by-step solution is like following a recipe. You start by understanding what is needed, proceed through the problem systematically, and complete each operation in order. This ensures accuracy and thoroughness.
In our task:

• We first understand that the problem involves evaluating the function for multiple integer values.
• We substitute each integer from \(-4\) to \(2\) into the function \( y = |x+1| \).
• Each step involves computing the absolute value for each substitution and recording the \( y \) value obtained.

Following each step precisely ensures that none of the calculations are missed and the final results are accurate.

Piecewise Functions

The concept of piecewise functions relates to how we interpret absolute value expressions. A piecewise function has different expressions based on the input value of \( x \). The absolute value function can be viewed piecewise because:

• If the input \( x+1 \geq 0 \), then \(|x+1| = x+1\).
• If the input \( x+1 < 0 \), then \(|x+1| = -(x+1) = -x-1\).

Understanding that the expression \( |x+1| \) behaves differently based on whether \( x+1 \) is non-negative or negative is key to correctly evaluating it. For the problem \( y = |x+1| \), you can apply this piecewise thinking to easily ascertain:- If \( x \geq -1 \), simply use \( x+1 \).- If \( x < -1 \), rewrite it as \(-x-1\).This approach simplifies understanding and calculating the function across a range of integers.