Problem 44 - 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. $$f(x)=\left\\{\begin{array}{rll}x & \text { if } & x<0 \\\\-x & \text { if } & x \geq 0\end{array}\right.$$

Step-by-Step Solution

Answer

The range of the piecewise function \(f(x) = \{\begin{array}{rll}x & \text { if } & x<0 \\-x & \text { if } & x \geq 0\end{array}\}\) is \((-\infty, 0]\).

Step 1: Graph the first part of the function

The script in this step will be to plot the function \(f(x) = x\) for all values where \(x < 0\). This will be a linear line with a slope of 1 and will be decreasing as you move towards the left on the x-axis. Make sure the points on the line are open circles where \(x = 0\) because the piecewise function does not include 0 in the range of this function.

Step 2: Graph the second part of the function

The following task will be to plot the function \(f(x) = -x\) for all values where \(x \geq 0\). This will be a linear line with a slope of -1 and will be increasing as you move towards the right on the x-axis. Make sure there is a closed circle on the graph at the origin (\(x = 0\)) because the piecewise function includes 0 in this part.

Step 3: Determine the range

Use the graph to determine the interval of y-values represented. For this function, the graph extends to all y-values \(\leq 0\) for all x-values. It means the range is \((-\infty, 0]\).

Key Concepts

Domain and Range

In mathematics, the **domain** refers to all possible input values (or x-values) that are allowed by a function. It essentially denotes where the function "lives" on the x-axis. For piecewise functions, the domain is comprised of the intervals that each piece or segment of the function covers. For this specific piecewise function, the domain is given as \((-\infty, \infty)\), meaning it includes every possible real number.

The **range** of the function represents all possible output values (or y-values) resulting from the function. In this exercise, the range is derived from analyzing the graph of the piecewise function. For the given function, each piece accounts for certain y-values. - For the first piece, spanning the region where \(x < 0\), the output values are negative and can extend indefinitely downward.- For the second piece, spanning \(x \geq 0\), the graph touches at \(y = 0\) but never higher.

Therefore, the overall range of this piecewise function is \((-\infty, 0]\). This indicates that the function's output values are any real number that is zero or less, representing the combined y-values of both function segments.

Graphing Piecewise Functions

Piecewise functions combine multiple sub-functions, each with its own distinct rule and interval. When graphing a piecewise function, it's helpful to:

• Break down the graph into its constituent parts by examining the specific functions defined over given intervals.
• Graph each part of the function individually, being very mindful of the conditions (such as <, ≤, >, ≥) that dictate where each piece applies.
• Use open or closed circles to clearly indicate whether end points are included or excluded in each interval.

In our exercise:- **First Piece**: Function is \(f(x) = x\) over \(x < 0\). It's a linear function with a positive slope (1). Plot a line extending leftward from zero, using an open circle to indicate exclusion of the point \((0,0)\).- **Second Piece**: Function is \(f(x) = -x\) over \(x \geq 0\). Graph a descending line from zero with a slope of -1. Use a closed circle at \((0,0)\) due to inclusion. This helps us know exactly how the pieces join or diverge on a graph.Following these steps helps create a smooth overall graph that accurately reflects the behavior of the piecewise function.

Linear Functions

**Linear functions** form the building blocks of various mathematical models. A linear function has a straightforward equation represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.- **Slope (m)**: It dictates the steepness of the line. A positive slope implies the line tilts upwards as you move along the x-axis, while a negative slope means the line tilts downwards.- **Y-intercept (b)**: It's the point where the line crosses the y-axis.

In a piecewise context, each sub-function is often linear, as seen in this exercise.- With \(f(x) = x\) for \(x < 0\), the slope is 1 (upward slope) with no shift along the y-axis since the y-intercept is 0.- For \(f(x) = -x\) where \(x \geq 0\), the slope is -1 (downward slope), also shifting through the origin (i.e., y-intercept is 0).Graphing a linear function accurately involves understanding these components, as each influences how the piece will plot on a graph. Linear functions form straight lines that can be effortlessly combined in piecewise functions to build more complex structures.