Problem 37 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=\left\\{\begin{aligned} x & \text { if } x<-3 \\ 2 & \text { if }-3 \leq x<1 \\\\-2 x+2 & \text { if } x \geq 1 \end{aligned}\right.\)

Step-by-Step Solution

Answer

Domain: \((- fty, fty)\); Range: \((-\infty, -3) \cup \{2\} \cup (0, \infty)\).

Step 1: Analyze the Function

The function is a piecewise function defined in three intervals based on the value of \(x\). Each piece of the function corresponds to a different expression: \(f(x) = x\) for \(x < -3\), \(f(x) = 2\) for \(-3 \leq x < 1\), and \(f(x) = -2x + 2\) for \(x \geq 1\).

Step 2: Determine the Domain

The function is defined for all real numbers since there are no restrictions on \(x\) in any of the intervals. Hence, the domain of \(f(x)\) is all real numbers, \((-\infty, \, \infty)\).

Step 3: Determine the Range by Analyzing Each Piece

For the first piece, \(f(x) = x\) for \(x < -3\), the outputs are all real numbers less than -3, i.e., \((-\infty, -3)\).For the second piece, \(f(x) = 2\) for \(-3 \leq x < 1\), the output is constantly 2.For the third piece, \(f(x) = -2x + 2\) for \(x \geq 1\), the outputs are decreasing starting from 0 (since \(-2(1) + 2 = 0\)), so it's \((-fty, 0]\).Thus, combining these sets, the range of \(f(x)\) is \((-\infty, -3) \cup \{2\} \cup (-\infty, 0]\), which simplifies to \((-\infty, -3) \cup \{2\} \cup (0, \infty)\).

Step 4: Graph Each Interval

Graph \(y = x\) for \(x < -3\) as a line extending to the left, stopping at \(-3\) with an open circle.Graph \(y = 2\) as a horizontal line from \(x = -3\) (closed dot) to \(x = 1\) (open circle).Graph \(y = -2x + 2\) starting at \(x = 1\) (closed dot) and extending to the right.

Key Concepts

Domain and Range of Piecewise Functions

In mathematics, the domain and range are fundamental tools for understanding functions, and they are especially crucial when dealing with piecewise functions. A piecewise function has different expressions based on different intervals of the input variable, often represented by \(x\). To fully understand these functions, you must first grasp the concepts of domain and range.

The **domain** of a function is the complete set of possible input values (\(x\) values) that will not result in any mathematical errors such as division by zero or the square root of a negative number. However, in the case of the exercise function \(f(x)\), it is expressed as a piecewise function with intervals that cover all real numbers, meaning our domain is

• All real numbers: \((-\infty, \, \infty)\)

Meanwhile, the **range** refers to the set of possible output values or the \(y\) values that the function can produce after applying the function rule over the domain. Each piece of the piecese function creates different outputs in specific intervals.

• For \(f(x) = x\) when \(x < -3\), the range is \((-\infty, -3)\).
• For \(f(x) = 2\) when \(-3 \leq x < 1\), the range is \(\{2\}\).
• Lastly, for \(f(x) = -2x + 2\) when \(x \geq 1\), it produces outputs that decrease starting from 0, giving the range of \((-\infty, 0]\).

The **complete range** is essentially the combination of all these intervals: \((-\infty, -3) \cup \{2\} \cup (0, \infty)\). Understanding the domain and range helps illustrate which inputs lead to specific outputs.

Graphing Piecewise Functions

Graphing piecewise functions offers visual insight into their behavior, distinctly showing how different expressions interact over a range of \(x\). To graph a piecewise function like \(f(x)\) from the exercise, it's key to handle each part separately, respecting its domain and range.

Begin with graphing each piece individually:

• Start with \(y = x\) for \(x < -3\). This part of the graph is a straight line extending to the left, stopping sharply at \(-3\) where you would place an open circle to indicate \(-3\) is not included.
• Next, graph \(y = 2\), a constant function, for the interval \(-3 \leq x < 1\). Draw a horizontal line from \(-3\) to \(1\). Use a closed dot at \(-3\) to show inclusion and use an open circle at \(1\) to show exclusion.
• Finally, graph \(y = -2x + 2\) for \(x \geq 1\). This is a line with a slope of \(-2\), starting at the point \( (1, 0) \), hence a closed dot at \(x = 1\) to show its inclusion, and drawn continuing to the right.

Assembling these segments on a single set of axes creates a complete picture of the piecewise function. These actions, along with understanding the breaks and reconnects at different \(x\)-values, highlight not only the function's complexity but its utility in modeling various situations.

Understanding Real Numbers in Function Contexts

Real numbers form the backbone of input values in mathematical functions, especially for piecewise definitions. They include every number along the number line, comprising the set of both rational numbers (like \( \frac{1}{2}\), \(5\), \(0\)) and irrational numbers (like \(\pi\), \(\sqrt{2}\)). Real numbers can be counted upon to complete a full description of the domain in many functional contexts.

In the realm of our exercise’s piecewise function, it's critical to remember that all pieces are defined over intervals of real numbers.

• The segment \(f(x) = x\) for \(x < -3\) involves real numbers extending infinitely in the negative direction.
• The constant value segment \(f(x) = 2\) guarantees that all real number inputs within \(-3\) and \(1\) map precisely to the real number 2.
• For \(f(x) = -2x + 2\) when \(x \geq 1\), all real numbers starting from \(x=1\) result in outputs stretching into \((-\infty, 0]\), depicting an expansive use of real inputs and outputs.

This feature of including all real numbers in the domain underlines the unrestricted continuity and completeness of functions, specifically in piecewise conditions. This provides confidence in the function's applicability to real-world problems where such conditions constantly fluctuate.