Problem 6 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=|3 x-2|\)

Step-by-Step Solution

Answer

Domain: (-∞, ∞), Range: [0, ∞)

Step 1: Understanding the Absolute Value Function

The function given is an absolute value function, which generally has a V-shaped graph. The absolute value function is defined as follows:\[f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}\]In this case, the function is \(f(x) = |3x - 2|\), meaning we need to solve the expression within the absolute value.

Step 2: Finding Critical Points

Set the expression within the absolute value to zero to find critical points where the graph changes direction: \[3x - 2 = 0\]Solving for \(x\), we add 2 to both sides:\[3x = 2\]Then, divide by 3:\[x = \frac{2}{3}\]This is the point where the graph changes direction and represents the vertex of the V shape.

Step 3: Drawing the Graph

To draw the graph, note that the graph is V-shaped and has a vertex at \(x = \frac{2}{3}\). Calculate some additional points:- For \(x = 0: f(0) = |3(0) - 2| = 2\)- For \(x = 1: f(1) = |3(1) - 2| = 1\)Plot these points and the vertex on a coordinate plane and draw the V, opening upwards.

Step 4: Identifying the Domain

The domain of the function \(f(x) = |3x - 2|\) is the set of all real numbers, since there is no restriction on the values \(x\) can take. So, the domain is:\[-\infty, \infty\].

Step 5: Identifying the Range

The range of the absolute value function is all non-negative real numbers because the absolute value is always zero or positive. Therefore, the range is:\[[0, \infty)\].

Key Concepts

Absolute Value Function

An absolute value function is characterized by its V-shaped graph. This distinctive shape is due to the nature of the absolute value operator itself. The absolute value of any number is its distance from zero on the number line, which means it is always non-negative. Mathematically, the absolute value function can be described as:

• \(f(x) = |x| = x\) when \(x \geq 0\)
• \(f(x) = |x| = -x\) when \(x < 0\)

For the function \(f(x) = |3x - 2|\), this means we are essentially looking at how the expression \(3x - 2\) behaves when the value is non-negative versus when it is negative. The graph illustrates this by flipping along the x-axis at the point where the expression inside the absolute value is zero, creating the V-shape.

Domain and Range

The concepts of domain and range are essential when discussing functions. The domain refers to all possible input values (the x-values) that the function can accept. For the absolute value function \(f(x) = |3x - 2|\), there are no restrictions on \(x\) because any real number is valid. Therefore, the domain is:

• Domain: \((-\infty, \infty)\)

On the other hand, the range refers to all possible output values (the y-values) that result from applying the function. Since the absolute value function always yields non-negative results, our range starts from zero and goes to infinity:

• Range: \([0, \infty)\)

This range makes sense in the context of the absolute value function because no output can ever be negative.

Critical Points

Critical points in the context of graphing an absolute value function are places where the graph changes direction. For the function \(f(x) = |3x - 2|\), the critical point occurs where the expression inside the absolute value equals zero. To find it, solve:\[3x - 2 = 0\]By solving this equation, we get:\[x = \frac{2}{3}\]This point, \(x = \frac{2}{3}\), is where the linear equations \(3x - 2\) and \(-3x + 2\) meet. It represents a sharp turn in the graph, forming the vertex of the V. Critical points are crucial because they often indicate local minimums or maximums in the graph.

Vertex of the Graph

The vertex of the graph of an absolute value function is the point where the graph makes its sharpest turn and resembles the base of the V-shape. For the function \(f(x) = |3x - 2|\), this occurs precisely at the critical point we discovered earlier:

• Vertex at \(x = \frac{2}{3}\)

To find the y-coordinate of the vertex, plug \(x = \frac{2}{3}\) back into the function:\[f(x) = |3(\frac{2}{3}) - 2| = |2 - 2| = 0\]Thus, the vertex of the graph is the point \((\frac{2}{3}, 0)\). The vertex not only marks the point of direction change but is also the lowest point on the graph for the function \(f(x) = |3x - 2|\). Understanding the vertex helps in accurately sketching and interpreting the function's behavior.