Problem 36 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=\left|x+\frac{1}{2}\right|\)

Step-by-Step Solution

Answer

Domain: all real numbers. Range: \([0, \infty)\).

Step 1: Understand the Function

The function is an absolute value function, given by \(f(x)=\left|x+\frac{1}{2}\right|\). This function represents a V-shaped graph, with its vertex shifting depending on the term inside the absolute value.

Step 2: Identify the Vertex

For \(f(x)=\left|x+\frac{1}{2}\right|\), set \(x+\frac{1}{2}=0\) to find the x-coordinate of the vertex. Solving this gives \(x=-\frac{1}{2}\). The vertex of the graph is \((-\frac{1}{2}, 0)\).

Step 3: Determine the Domain and Range

For the function \(f(x)=\left|x+\frac{1}{2}\right|\), the domain is all real numbers since there are no restrictions on \(x\). The range is all non-negative real numbers, \([0, \infty)\), because an absolute value cannot be negative.

Step 4: Plot Key Points

Choose and evaluate key points around the vertex to understand the graph's shape. Start by evaluating \(f(x)\) at a few points: \(f(-2)=1.5\), \(f(-1)=0.5\), and \(f(0)=0.5\).

Step 5: Sketch the Graph

Using the vertex and key points, draw the V-shaped graph. Begin at the vertex \((-\frac{1}{2}, 0)\). Plot the points \((-2, 1.5)\), \((-1, 0.5)\), and \((0, 0.5)\), and draw lines through these points to visualize the graph's structure, forming a V shape that opens upwards.

Key Concepts

Graphing Functions

The process of graphing functions begins with understanding the nature of the function itself. For absolute value functions such as \(f(x)=\left|x+\frac{1}{2}\right|\), this first involves recognizing its V-shaped structure. The graph of an absolute value function always opens upwards and shows symmetry about the vertical line through its vertex.

To start graphing \(f(x)=\left|x+\frac{1}{2}\right|\), we identify the vertex. The vertex is the point where the expression inside the absolute value equals zero. In this case, \(x+\frac{1}{2}=0\). Solving this equation gives the vertex at \(x=-\frac{1}{2}\). Hence, the vertex is \((-\frac{1}{2}, 0)\).

It's crucial to plot this point on your graph first. From here, determine a few key points on either side of the vertex to help illustrate the V-shape of the absolute value function. Evaluating the function at these points, like \(x = -2\), \(x = -1\), and \(x = 0\), allows you to see how the graph behaves on either side of the vertex. When these points are plotted, and lines are drawn through them, the complete V-shaped graph of the function emerges.

Domain and Range

Understanding the domain and range of a function gives you a sense of the possible values the function can take.

For the function \(f(x)=\left|x+\frac{1}{2}\right|\), the domain refers to all the permissible values of \(x\). Since there are no restrictions on \(x\) in this absolute value function, the domain consists of all real numbers, noted as \((-\infty, \infty)\). This means you can input any real number into the function, and it will produce a result.

The range focuses on the possible output values. With absolute value functions, the results are always non-negative, since the absolute value operator itself cannot produce negative results. Therefore, the range for our function is \([0, \infty)\), representing all real numbers starting from zero and extending to positive infinity.

Vertex of a Function

The vertex is one of the most important features of an absolute value function. It is the point where the direction of the graph changes, known as the "corner" of the V. Understanding the vertex helps in sketching the graph and analyzing the function's behavior.

For the function \(f(x)=\left|x+\frac{1}{2}\right|\), the vertex is found by setting the expression inside the absolute value equal to zero. Solving \(x+\frac{1}{2}=0\) shows that \(x=-\frac{1}{2}\). This makes the vertex \((-\frac{1}{2}, 0)\).

Knowing the vertex gives us critical insight into where the function reaches its minimum point, at \(f(x)=0\). From this point, the graph extends upwards in both directions. Locating the vertex first is a practical step for beginners because it acts as an anchor that ensures the function is correctly represented when graphed.