Problem 40 - Algebra A Combined Function

Question

Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. $$ h(x)=\left\\{\begin{array}{rlr} x+2 & \text { if } & x<1 \\ 2 x-1 & \text { if } & x \geq 1 \end{array}\right. $$

Step-by-Step Solution

Answer

The domain and range of the function are both all real numbers, \((-\infty, \infty)\).

Step 1: Analyze the first piece

The function is given by two expressions that apply to different parts of the domain. The first piece, \( x+2 \), is defined for \( x < 1 \). This is a linear equation with a slope of 1 and a y-intercept at 2.

Step 2: Graph the first piece

To graph \( x+2 \) for \( x < 1 \), start by plotting a point at (0, 2), which is the y-intercept. The line has a slope of 1, meaning it rises one unit for each unit it moves to the right. Since \( x < 1 \), draw the line to the left of \( x = 1 \) and use an open circle at \( (1, 3) \) to indicate that this point is not included.

Step 3: Analyze the second piece

The second piece is \( 2x - 1 \) and it applies when \( x \geq 1 \). This is another linear equation with a slope of 2 and a y-intercept at -1.

Step 4: Graph the second piece

For \( 2x - 1 \), start by plotting a point at \( (1, 1) \) since when \( x = 1 \), \( 2(1) - 1 = 1 \). This graph includes the point (1, 1), so use a closed circle there. Continue the line to the right with the slope of 2.

Step 5: Determine the domain of the function

The domain of the function is the set of all possible \( x \)-values the function can take. Since the first piece covers all \( x < 1 \) and the second piece covers \( x \geq 1 \), the domain is all real numbers \(( -\infty, \infty)\).

Step 6: Determine the range of the function

The range is the set of all possible \( y \)-values the function can take. The first piece \( y=x+2\) starts from \( y \) approaching -\(\infty\) to just below 3. The second piece, \( y=2x-1 \), starts at \( y = 1 \) and goes to \( \infty \). Hence, the range is \((-\infty, \infty)\) as well.

Key Concepts

Understanding Domain and Range

In mathematics, the domain and range are fundamental concepts when dealing with functions. The **domain** is all about the x-values that can input into a function. It tells us where our function 'lives' on the x-axis.
For example, in this piecewise function, it includes values like all the fitness levels we could see. For the given exercise:

• The first part of the function, \( x+2 \), is applied when \( x<1 \).
• The second part, \( 2x-1 \), occurs for \( x\geq1 \).
• Together, this means our domain covers all real numbers, as we can use any x-value.

The **range**, however, focuses on all the y-values the function produces. It's the output, or what shows up on the y-axis.
Here, the range goes from very low on the y-axis as \( x \to -\infty\) for the first piece, up to infinity for the second piece. So, our range also includes all real numbers.
Together, defining the domain and range gives us a complete picture of where our function operates and what results it yields.

Exploring Linear Equations

Linear equations are perhaps the simplest type of equations one can come across. They graph as straight lines in a coordinate plane and take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In our piecewise-defined function, we encounter two linear equations:

• \(x + 2\) with a slope of 1 and y-intercept 2.
• \(2x - 1\) with a slope of 2 and y-intercept -1.

The slope \(m\) indicates how steep the line is. Since slope is rise over run, it tells us how much the line goes up or down as \(x\) increases. For instance, a slope of 1 signifies the line rises one unit up for each x increase by one unit.
The y-intercept \(b\) is where the line hits the y-axis. It's what y equals when x is zero, offering a starting point when graphing.
Understanding these characteristics helps in accurately drawing and interpreting linear graphs.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting separate line segments or curves confined to specific portions of the x-axis. In our exercise, the function switches rules at \(x = 1\).
Here’s how you can graph this type of function:1. **Identify each piece of the function:** We have two linear pieces based on whether x is less than or equal to 1.2. **Plot points:** Start by plotting the y-intercept of each line within its specified domain. For example, point \((0, 2)\) for \(x + 2\) and \((1, 1)\) for \(2x - 1\).3. **Draw lines:** Use the slope to plot further points and draw the lines for each piece. Remember: - For \(x < 1\): Make an open circle at \((1, 3)\) to show it’s not included. - For \(x \geq 1\): Use a closed circle at \((1, 1)\) indicating inclusion.This method ensures a clear and precise visual representation, allowing you to identify changes and behavior across different x values.