Problem 37 - Algebra A Combined Function

Question

Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. $$ h(x)=\left\\{\begin{array}{lll} 5 x-5 & \text { if } & x<2 \\ -x+3 & \text { if } & x \geq 2 \end{array}\right. $$

Step-by-Step Solution

Answer

Domain: All real numbers. Range: \((-\infty, 5) \cup [1, \infty)\).

Step 1: Understand Piecewise Functions

Piecewise functions have different expressions based on different intervals of the input variable. The given function has two expressions: 1. \(5x-5\) for \(x < 2\)2. \(-x+3\) for \(x \geq 2\). Each part needs to be graphed on its respective interval.

Step 2: Graphing the First Piece

Graph the line \(y = 5x - 5\) for values where \(x < 2\). This is a line with slope 5 and y-intercept -5. Since this is defined only for \(x < 2\), the line will end at \(x = 2\) with an open circle at the point \((2, 5(2)-5) = (2, 5)\) since it doesn't include \(x = 2\). Draw this line starting from the y-intercept and moving towards the point \((2,5)\) without including that point.

Step 3: Graphing the Second Piece

Graph the line \(y = -x + 3\) for \(x \geq 2\). This is a line with a slope of -1 and y-intercept 3. Start from the point \((2, 3)\) since \(x = 2\) is included in this piece (use a filled circle) and draw the line extending rightwards indefinitely.

Step 4: Identify the Domain

The domain is the set of all input values for which the function is defined. Both pieces together cover all real numbers, so the domain of \(h(x)\) is all real numbers: \((-\infty, \infty)\).

Step 5: Identify the Range

The range is the set of all possible output values of the function. From the graph, for \(x < 2\), the output values are from \(-\infty\) up to, but not including, 5. For \(x \geq 2\), the output starts at 1 (when \(x = 2\) gives \(y = 1\)) and decreases indefinitely. Therefore, the range is \((-\infty, 5) \cup [1, \infty)\).

Key Concepts

Graphing Functions

Graphing piecewise functions involves plotting each defined expression over their specified intervals. In our example, we have two separate linear expressions depending on the value of the variable \(x\). For the function

• \(5x-5\) is plotted when \(x < 2\), and
• \(-x+3\) is plotted when \(x \geq 2\).

To correctly graph these functions, you need to start by identifying the pertinent details of each linear piece.When graphing \(5x-5\) for \(x < 2\), we focus on a line with a slope of 5 and a y-intercept at -5. On the graph, this piece appears as a line extending from the y-axis to slightly before \(x = 2\). The line ends at point \((2,5)\), noted by an open circle to indicate that this point is not part of the graph. For the piece \(-x+3\) with \(x \geq 2\), graph the line starting at point \((2, 1)\) with a filled circle. This signifies \(x = 2\) and all greater values are included. The line continues infinitely rightward as the slope of -1 carries it downward. By carefully graphing these segments, you create an accurate visual representation of the function.

Domain and Range

The domain of a function is the complete set of possible input values (x-values) for which the function is defined, while the range is the complete set of possible output values (y-values).For our piecewise function, the domain encompasses all real numbers because both linear expressions together cover every possible \(x\)-value without any gaps. Thus, the domain is \((-\infty, \infty)\).The range is slightly more complex due to the division of \(x\) values across the two expressions:

• For \(x < 2\) with \(y = 5x - 5\), outputs go up to just under 5, effectively making it \((-\infty, 5)\).
• Meanwhile, \(x \geq 2\) using \(y = -x+3\) produces outputs starting at 1 and moving downward towards negative infinity, yielding the range \([1, \infty)\).

Thus, the overall range when combined is \((-\infty, 5) \cup [1, \infty)\), ensuring complete coverage for every output value the graph can attain.

Linear Functions

Linear functions are equations that graph as straight lines. They play a crucial role in this piecewise function as each segment is linear. These functions follow the general form \(y = mx + c\), where:

• \(m\) is the slope, indicating steepness or rate of change, and
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

In the function \(h(x)\), the first expression \(5x-5\) implies:- **Slope (m):** 5, denoting a steep rise.- **Y-intercept (c):** -5, crossing the y-axis downward.For the second expression \(-x+3\):- **Slope (m):** -1, indicating a mild descent.- **Y-intercept (c):** 3, showing a crossing of the y-axis above the origin.Understanding the role of slope and y-intercept helps predict how each piece behaves within its domain, shaping the overall graph and facilitating accurate plotting and analysis of the piecewise function.