Problem 64 - Precalculus Essentials

Question

The domain of each piecewise function \(i s(-\infty, \infty)\) 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 \(-5 \leq y < \infty\). For \(x < -3\), the function \(f(x) = x + 2\) applies and for \(x \geq -3\), the function \(f(x) = x - 2\) applies. Both parts of the piecewise function are graphed according to their respective conditions to determine the range.

Step 1: Graphing the Function

Start by graphing both parts of the piecewise function separately. For the first equation \(x + 2\) which is applicable for \(x < -3\), prepare a set of points like (-4, -2), (-5, -3), etc., and plot them. For the second equation \(x - 2\) which is applicable for \(x \geq -3\), prepare a set of points like (-3, -5), (-2, 0), etc., and plot them.

Step 2: Consider the Conditions

Keep in mind the applicable conditions for each equation while plotting. The function \(x + 2\) applies only when \(x < -3\) while \(x - 2\) applies when \(x \geq -3\). So when \(x \geq -3\), remove the parts of first equation's graph and when \(x < -3\), remove the parts of second equation's graph.

Step 3: Determine the Range from Graph

After we complete the graph, we will look at the y-values of the plotted points. The smallest y-value that belongs to the function is -5 and there is no upper limit, therefore the range of this function is \(-5 \leq y < \infty\).

Key Concepts

Function Domain

When dealing with piecewise functions, understanding the function's domain is essential. The domain refers to all the possible x-values for which the function is defined. A piecewise function consists of different expressions over specific intervals of the domain.

For example, in the function provided:

• The expression \(x+2\) is valid for \(x < -3\).
• The expression \(x-2\) is valid for \(x \geq -3\).

The provided interval \((-\infty, \infty)\) indicates that the function is defined for all real numbers. Yet, it is split into two pieces, each applicable under different conditions based on the x-value. This means that you need to focus on the conditions assigned to each piece to understand where exactly each part of the function applies.

Remember, correctly identifying the domain allows you to appropriately set up conditions for graphing the function and analyzing its behavior.

Function Range

While the domain deals with x-values, the range of a function is about its y-values. It's the set of possible outputs your function can produce. To find the range, especially for piecewise functions, you often need to graph the function to see the extent of y-values it covers.

The piecewise function \(f(x)=\left\{\begin{array}{ll} x+2 & \text{if} \ x<-3 \ x-2 & \text{if} \ x \geq-3 \end{array}\right.\) has two separate linear components. Each will contribute to different parts of the range based on its own rule.

From the graph, the y-values start at -5 and can increase towards infinity. The key points here are:

• The smallest y-value is -5, when \(x = -3\).
• Given \(y = x + 2\) is unbounded, the range extends to positive infinity.

This leads us to conclude that the range of the function is \([-5, \infty)\), meaning any y-value greater than or equal to -5 could be a valid output.

Graphing Functions

Graphing piecewise functions involves a step-by-step approach to ensure each segment is plotted under the correct conditions. Let's break this down:

**Plotting each piece separately:** Start by identifying and plotting points for each piece of the function with respect to its specified condition:

• For \(x+2\) (when \(x < -3\)), plot points such as (-4, -2), (-5, -3), etc.
• For \(x-2\) (when \(x \geq -3\)), points can be (-3, -5), (-2, 0), etc.

**Consider the conditions:** It's crucial to remember the conditions that dictate for which x-values each piece of the function is valid. Only include those parts of the graph that adhere to these conditions.

Upon completing the plotting:

• Observe the graph to infer about the function's behavior.
• Notice any discontinuities or overlaps based on the conditions.

Graphing piecewise functions manually might seem complex at first, but with the right steps and careful attention to the intervals over which each piece is defined, it becomes manageable. The completed graph will visualize both the function's domain and its range, offering insights into how the function behaves across its domain.