Problem 22 - Algebra 2

Question

Graph each function. Identify the domain and range. \(g(x)=|x|+3\)

Step-by-Step Solution

Answer

The domain is \((-\infty, \infty)\); the range is \([3, \infty )\).

Step 1: Understand the Function

The function given is \( g(x) = |x| + 3 \). This represents a vertical shift of the basic absolute value function \( f(x) = |x| \) upwards by 3 units.

Step 2: Graph the Basic Function

First, graph the basic absolute value function \( f(x) = |x| \). This graph is a V-shaped graph with the vertex at the origin (0,0), symmetric with respect to the y-axis.

Step 3: Apply the Vertical Shift

Shift the entire graph of \( f(x) = |x| \) three units upward to account for the '+3' in \( g(x) = |x| + 3 \). Now, the vertex of the graph is at (0,3), and the graph maintains its V-shape and symmetry.

Step 4: Identify the Domain

The domain of the function \( g(x) = |x| + 3 \) is the set of all possible input values (x-values). Since absolute value functions are defined for all real numbers, the domain is \( (-\infty, \infty) \).

Step 5: Identify the Range

The range of the function \( g(x) = |x| + 3 \) is the set of all possible output values (y-values). Since the lowest point of the graph is at \( y = 3 \) and it extends upwards indefinitely, the range is \( [3, \infty) \).

Key Concepts

Domain and Range

When we talk about a function's domain, we are referring to all the possible input values (or x-values) that can be plugged into the function without breaking it. For the function \( g(x) = |x| + 3 \), the absolute value component \( |x| \) is defined for all real numbers. Therefore, the domain of the function is simply all real numbers, which we express as \( (-\infty, \infty) \).

On the other hand, the range is about the possible output values (y-values) the function can produce. Since the function takes the basic V-shape of \( f(x) = |x| \) and moves it up by 3 units, the bottommost point on the graph is at \( y = 3 \). As the graph extends infinitely upwards, the lowest value in its range is 3, and there is no upper limit. Thus, the range is \( [3, \infty) \). This means any value from 3 upwards could be an output of the function.

Vertical Transformations

Vertical transformations involve moving a graph up or down on a coordinate plane. For our function \( g(x) = |x| + 3 \), the transformation is a vertical shift. To understand it, visualize moving the whole graph of \( f(x) = |x| \) upwards by 3 units.

Here are a few key points about how this affects the graph:

• The vertex moves from its original position at the origin (0,0) to a new position at (0,3).
• The V-shape of the graph does not change. It remains symmetric about the y-axis.
• Every point on the original \( f(x) = |x| \) graph is effectively "lifted" three units higher on the y-axis.

Understanding vertical transformations helps in quickly sketching transformed graphs without having to plot them from scratch every time.

Absolute Value Function

Absolute value functions are known for their distinctive V-shaped graphs. The simplest form, \( f(x) = |x| \), has a vertex at the origin (0,0). The graph opens upwards and is perfectly symmetrical about the y-axis. It reflects the idea that absolute value is the "distance" of a number from zero, always non-negative.

Whenever you add or subtract a constant to \( |x| \), such as in \( g(x) = |x| + 3 \), you are introducing what we call a vertical transformation. This shifts the entire graph up or down without altering the shape of the graph. It retains its V-form and symmetry around the y-axis.

An absolute value function can also be used to model various real-world situations, such as measuring distances irrespective of direction.

Function Graphing

Graphing functions allows us to visually interpret relationships between variables. For the function \( g(x) = |x| + 3 \), we start by graphing the basic absolute value function \( f(x) = |x| \). This involves plotting points that create a V-shape centered on the origin. The graph is symmetric about the y-axis, with each side extending towards infinity.

After plotting \( f(x) = |x| \), we apply the given transformations. In this case, add 3 to every y-value, moving the entire graph up by 3 units. The vertex of the graph moves to (0,3), maintaining the symmetry and shape of a V.

Points that can help in graphing include:

• Vertex: (0,3)
• Points like (1,4) and (2,5) that lie along the V's arms

Visualizing the graph aids in understanding the behavior of absolute value functions and how transformations modify their position.