Problem 46 - College Algebra

Question

The domain of each piecewise function is \((-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{lll}1 x & \text { if } & x \leq 0 \\\3 & \text { if } & x>0\end{array}\right.$$

Step-by-Step Solution

Answer

The range of the function \(f(x)\) is \((-\infty, 0] \cup \{3\}\)

Step 1: Understand the pieces of the function

The function \(f(x)\) has been defined differently for different sections of its domain. When \(x \leq 0\), \(f(x) = x\). This represents a straight line passing through the origin with a slope of 1. When \(x > 0\), \(f(x) = 3\). This represents a horizontal line at \(y = 3\).

Step 2: Graph the function

To graph this function, begin by drawing the y-axis and x-axis. Mark the point where \(x = 0\) on the x-axis. For \(x \leq 0\), the function is a straight line passing through the origin with a slope of 1. Draw this on the graph. For \(x > 0\), the function is constant at \(y = 3\). Draw a horizontal line starting from \(x = 0\) at \(y = 3\). To indicate that the value at \(x = 0\) belongs to the line \(f(x) = x\), use a filled-in dot at the point where the line \(f(x) = x\) intersects the y-axis (0,0). Similarly, use an empty dot at the point (0,3) on the line \(f(x) = 3\) to indicate that the value at \(x = 0\) does not belong to the line \(f(x) = 3\).

Step 3: Determine the function's range from the graph

The range of a function is the set of all possible output values (y-values). Looking at the graph of the function, it can be seen that as \(x\) ranges from \(-\infty\) to 0, \(f(x)\) ranges from \(-\infty\) to 0. As \(x\) ranges from 0 to \(\infty\), \(f(x)\) equals 3. Therefore, the range of \(f(x)\) is \((-\infty, 0] \cup \{3\}\).

Key Concepts

Domains and Ranges

Understanding domains and ranges is crucial for grasping how functions behave. The **domain** is all the possible input values, or **x-values**, that a function can accept. In the exercise, the piecewise function has a domain of \((-fty, \infty)\), meaning it accepts any real number as an input. This makes sense because the function has no restrictions like division by zero or taking the square root of a negative number.

The **range**, on the other hand, is all possible output values, or **y-values**, the function can produce. When you graph the piecewise function given, you notice that for all *x* less than or equal to 0, the function outputs y-values that range from \((-fty, 0]\). This is because as *x* decreases to \(-fty\), so do the *y* values. Conversely, for *x* greater than 0, the function outputs a constant value *y* = 3, giving us the discrete value **3** in the range.

Thus, combining these, the full range is \((-fty, 0] \cup \{3\}\). Always check the graph to see which y-values are hit, as it visually confirms the range.

Graphing Functions

Graphing piecewise functions involves understanding each "piece" of the function separately. Begin by plotting the axes. Then, sketch the different graphical sections overlaid on these axes based on the conditions given in the piecewise definition. This helps visually analyze and understand the behavior of the function across different segments of its domain.

For the function provided, let's break it into its parts:

• **For** \((x \leq 0)\), the function behaves like \(f(x) = x\), a straight line with a slope of 1 passing through the origin. Plot this section for negative x-values and include the point (0,0) as it is within this piece's range.
• **For** \((x > 0)\), \(f(x) = 3\) implies a horizontal line at y = 3 for all positive x-values. Indicate this with an open circle at (0,3) since x = 0 doesn't include this line.

By meticulously plotting each segment and the transition points (like open and closed circles), understanding the whole function becomes a breeze.

Function Notation

Function notation is a way to express how each input value maps to an output value in a function. It often looks like \(f(x)\), where \(f\) denotes the function's name, and \(x\) the input value. This notation is essential for distinguishing between different functions and understanding their behavior.

In our example, the function \(f(x)\) is defined piecewise, which means its definition changes based on the input value range (the domain conditions). This is shown by the curly braces and the specific conditions attached to each portion of the function:

• \