Problem 48 - College Algebra Essentials

Question

the domain of each piecewise function is \((-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$ f(x)=\left\\{\begin{array}{lll} x+2 & \text { if } & x<-3 \\ x-2 & \text { if } & x \geq-3 \end{array}\right. $$

Step-by-Step Solution

Answer

The range of the function is \((-\infty, \infty)\)

Step 1: Graph for x < -3

To graph the first piece 'x+2', first draw a number line and consider values of x less than -3. Take a point from this range (for example, x=-4), substitute this value in 'x+2' which gives us -4+2=-2. Therefore, at x=-4, y=-2. The graph would be a straight line and stops at x=-3 without including the point at x=-3 (since x is strictly less than -3).

Step 2: Graph for x >= -3

The second piece 'x-2' will be graphed for values greater than or equal to -3. So, take a point from this range (for example, x=-2), and substitute this value in 'x-2'. This give us -2-2=-4. So, at x=-2, y=-4. This graph is also a straight line but unlike the first piece, it includes the point at x=-3 (since x>=-3)

Step 3: Final Graph

The final graph is a combination of the graphs from Step 1 and Step 2. On the same graph, represent the two lines.

Step 4: Determine Function's Range

Observing the graph, it can be seen that the y-values (function's range) cover all real numbers, so the range of the function is \((-\infty, \infty)\)

Key Concepts

Domain and Range

Understanding the domain and range of a piecewise function is essential. The **domain** is the set of all possible input values (x-values) that a function can accept. For our piecewise function, the domain is \((-\infty, \infty)\), which means it can accept any real number as input. This is common with piecewise functions when there are no restrictions on x.
The **range** is the set of possible output values (y-values) the function can produce. In our exercise, after graphing the function, we find the range to be \((-\infty, \infty)\) as well. This implies that the graph extends infinitely in both upward and downward directions. Observing the graph shows how values shift smoothly between the two linear components, supporting the conclusion that all real y-values are covered.

Graphing Functions

Graphing a piecewise function isn't too different from graphing any other function, except it requires extra care to consider each "piece" separately. In our piecewise function, we have two sections: \(x+2\) for \(x < -3\) and \(x-2\) for \(x \geq -3\).
To start, identify the different pieces and their respective conditions. To graph **\(x+2\)**:

• Choose x-values less than -3, like -4 or -5.
• Calculate y-values using the formula (e.g., \(-4+2 = -2\)).
• Plot these points on the graph and draw the line, ensuring it stops just before x reaches -3.

For **\(x-2\)**:

• Select x-values greater than or equal to -3, such as -3 or -2.
• Use the formula to find y-values (e.g., \(-3-2 = -5\), \(-2-2 = -4\)).
• Include the point at x = -3, where the line continues for larger x-values.

Both pieces graph as straight lines, and combined, they form the full graph of the function.

Linear Functions

The concept of linear functions is fundamental when dealing with piecewise functions. Each "piece" of a piecewise function is usually a linear function or comprised of linear segments.
A **linear function** is any function that can be graphed as a straight line. For instance, \(x+2\) and \(x-2\) are both linear. They follow the format \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• **Slope (m):** This determines the steepness of the line. Both functions in our exercise have a slope of 1, meaning they increase at a consistent rate.
• **Y-intercept (b):** This is where the line crosses the y-axis. For \(x+2\), the y-intercept is 2. For \(x-2\), it's -2.

Understanding these features allows us to predict and verify the shape and direction of each line before graphing, simplifying the process of working with piecewise functions.