Problem 9 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ \phi=\\{(x, y)|y=| 3 x+2 \mid\\} $$

Step-by-Step Solution

Answer

Domain: \( (-\infty, \infty) \). Range: \( [0, \infty) \). The graph is a 'V' shape.

Step 1: Understand the function

The function given is \( \phi = \{(x, y) | y = |3x + 2|\}\). This means the function maps each x-value to the corresponding y-value which is the absolute value of \( 3x+2\).

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (x-values). Since the function \( y = |3x + 2| \) is defined for all real numbers, the domain is all real numbers. In interval notation, this is \( (-\infty, \infty) \).

Step 3: Determine the Range

The range of a function is the set of all possible output values (y-values). For the absolute value function \( y = |3x + 2| \), the output can never be negative. Therefore, the range is all non-negative real numbers, i.e., \( [0, \infty) \).

Step 4: Sketch the Graph

To sketch \( y = |3x + 2| \), remember that absolute value functions create a 'V' shape. \( 3x + 2 = 0 \) when \( x = -\frac{2}{3} \), and at this point the graph changes direction. On the left of \( x = -\frac{2}{3} \) the function is decreasing, and on the right, it is increasing. Plot a few points to get an accurate graph, for example: \( (-2, 4)\) and \( (0, 2)\).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It is denoted as \(|x|\), and it represents the distance of a number on the number line from zero. It always gives a non-negative value. For any real number x:

• If x is positive or zero, then \(|x| = x\).
• If x is negative, then \(|x| = -x\).

In our exercise, the function is given by \(\rightarrow) y = |3x + 2|\). This means that whatever the value of \(\rightarrow)3x + 2\) is, the output will always be its non-negative value. This property creates a 'V' shape in the graph.

Graph Sketching

Graphing functions is a crucial skill for visualizing mathematical relationships. For the absolute value function \(\rightarrow) y = |3x + 2|\), the graph transforms around the point where \(\rightarrow)3x + 2 = 0\). This transformation point, known as the vertex, is where the function changes direction. In this case:

• First, set \(\rightarrow)3x + 2 = 0 \rightarrow x = -\frac{2}{3} \).

To sketch the graph, follow these steps:

• Identify the vertex. Here, it's \(\rightarrow)(-\frac{2}{3}, 0)\).
• Determine how the function behaves on either side of this vertex. Calculate a few points on the graph, for instance: \(\rightarrow)(-2, 4)\) and \(\rightarrow)(0, 2)\).
• Draw the graph noting the 'V' shape centered at the vertex with different slopes on either side.

The slope changes at \(\rightarrow) x = -\frac{2}{3} \). On the left, the function decreases, and on the right, it increases.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is used to denote domains and ranges of functions compactly and precisely.
Here are some basics of interval notation:

• The round bracket \(\rightarrow()\) denotes that the endpoint is not included (open interval).
• The square bracket \(\rightarrow[]\) means the endpoint is included (closed interval).
• For example, \(\rightarrow)(-3, 5]\) represents all numbers greater than -3 and up to 5, including 5 but not -3.

In the exercise, the domain of \(\rightarrow) y = |3x + 2|\) is all real numbers because the absolute value function is defined for all \(\rightarrow)x\) values. Thus, the domain in interval notation is \(\rightarrow)(-\backslashinfty, \backslashinfty)\). The range of the function is all non-negative real numbers because absolute values are never negative. Therefore, the range in interval notation is \(\rightarrow)[0, \backslashinfty)\).