Problem 45 - 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. $$f(x)=\left\\{\begin{array}{lll}2 x & \text { if } & x \leq 0 \\\2 & \text { if } & x>0\end{array}\right.$$

Step-by-Step Solution

Answer

The range of the piecewise function is \(-\infty \leq y \leq 2\)

Step 1: Identify Parts of the Piecewise Function

The piecewise function provided has two key parts: 1. \(f(x) = 2x\) for \(x \leq 0\). This indicates that for every \(x\), which is less than or equal to 0, we multiply x by 2 to get the y-value. Since there is no restriction on \(x = 0\), this part of the function is defined for all \(x \leq 0\). 2. \(f(x) = 2\) for \(x > 0\). This indicates that for every \(x\), which is greater than 0, the y-value will always be 2, regardless of the value of \(x\).

Step 2: Graphing the Piecewise Function

To graph the piecewise function, we can imagine splitting it into two parts, one for each segment of the domain. 1. For the first part, since \(f(x) = 2x\) for \(x \leq 0\), that's a straight line with a slope of 2, passing through origin (where x = 0). Here, you plot the line on the left-hand side of the y-axis (including the origin). 2. For the second segment where \(f(x) = 2\) for \(x > 0\), this indicates that \(y\) always equals 2, regardless of the value of \(x\), for every \(x > 0\). So, this is a horizontal line passing through 2 on the y-axis. Join these two individual graphs together at the point (0,0) for a complete graph of this piecewise function.

Step 3: Determine the Range of the Function

The range of a function is the set of all possible output values (y-values). Looking at our graph, we can see that when \(x \leq 0\), \(f(x)\) is equal to \(2x\), and considering our domain is \(-\infty\) to \(\infty\), the y-values would spread from \(-\infty\) up to \(0\). When \(x >0\), \(f(x)\) is always \(2\). So, the range of this function is \(-\infty \leq y \leq 2\).

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial for interpreting graphs. The domain of a function refers to all the possible input values (x-values) that the function can accept. For the piecewise function provided, the domain is (-\infty, \infty), meaning it includes all real numbers.
The range is the set of all possible output values (y-values). In this piecewise function:

• For \(x \leq 0\), the function is \(f(x) = 2x\), which ranges from negative infinity up to 0.
• For \(x > 0\), the function is \(f(x) = 2\), a constant value.Thus, the full range is from \(-\infty\) to 2.

Understanding these concepts helps in effectively interpreting and drawing the graph.

Graphing Functions

Graphing piecewise functions involves plotting each segment separately based on its defined conditions. Here’s how you can graph the given function step-by-step.
Firstly, identify the segments of the graph:

• For \(x \leq 0\), graph \(f(x) = 2x\). This is a linear function with a slope of 2 and it includes the point (0,0).
• For \(x > 0\), plot the horizontal line \(f(x) = 2\). This line begins just after \(x = 0\) and remains at a constant y-value of 2.

When graphing these parts together, ensure you use a filled point at (0,0) where the line from \(f(x) = 2x\) stops, and an open point at (0,2) where the constant part starts. This visually distinguishes the transition on the graph.

Linear Functions

Linear functions are the backbone of piecewise functions, especially noticeable in segments where you deal with straight lines. A linear function is expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In the piecewise function:

• The part \(f(x) = 2x\) has a slope (m) of 2 and a y-intercept of 0, meaning the line passes through the origin.
• For all \(x \leq 0\), this line continues diagonally across the negative x-values.

Linear functions are easy to graph because they form a straight line, making calculations straightforward. Understanding the slope and intercept gives insight into how the graph behaves, especially where changes occur in piecewise functions.