Problem 11 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 34, the function is the set of all ordered pairs \((x, y)\) satisfying the given equation. Find the domain and range of the function, and draw a sketch of the graph of the function. $$ G: y=\left\\{\begin{aligned} -2 & \text { if } x \leq 3 \\ 2 & \text { if } 3

Step-by-Step Solution

Answer

Domain: \( (-\infty, \infty) \); Range: \( \{-2, 2\} \); Sketch: Horizontal lines at \( y = -2 \) and \( y = 2 \) with circles at \( x = 3 \).

Step 1 - Analyze the Equation

The function is given in a piecewise manner: For \( x \leq 3 \), \( y = -2 \). For \( x > 3 \), \( y = 2 \).

Step 2 - Determine the Domain

The domain of the function is the set of all possible values of \( x \). Since there are no restrictions on \( x \), the domain is: \( (-\infty, \infty) \).

Step 3 - Determine the Range

The range of the function is the set of all possible values of \( y \). The function can only take the values -2 and 2, hence the range is: \( \{-2, 2\} \).

Step 4 - Draw the Sketch

Sketch the graph by drawing a horizontal line at \( y = -2 \) for \( x \leq 3 \) and another horizontal line at \( y = 2 \) for \( x > 3 \). Place a filled circle at \( (3, -2) \) to indicate \( y = -2 \) is included for \( x = 3 \) and an open circle at \( (3, 2) \) to indicate \( y = 2 \) starts right after \( x = 3 \).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial while analyzing functions. The domain represents all possible input values (normally the x-values) for which the function is defined. In the case of a piecewise function like our example:
\[ y= \left\{ \begin{aligned} -2 & \text{ if } x \leq 3 \ 2 & \text{ if } x > 3 \end{aligned}\right. \]
We have two parts: one for \(x \leq 3\) and one for \(x > 3\). Since there are no restrictions given, the function is defined for every real number. This means that every x-value on the number line works in the function.
Therefore, the domain is \((-fty, fty)\). This notation expresses that the domain consists of all real numbers from negative infinity to positive infinity.

Range of a Function

The range of a function refers to all possible output values (y-values) that the function can produce. For our piecewise function:
\[ y= \left\{ \begin{aligned} -2 & \text{ if } x \leq 3 \ 2 & \text{ if } x > 3 \end{aligned}\right. \]
We observe that the function can only output specific y-values: either -2 or 2. Let's break it down:

• For any \( x \leq 3 \), the output is \( y = -2 \).
• For any \( x > 3 \), the output is \( y = 2 \).

So, the range does not contain any other values apart from -2 and 2. Clearly, our range is limited to just these two values. Therefore, we write the range as a set of these values: \( \{-2, 2\} \).

Graphing Piecewise Functions

Graphing piecewise functions can seem tricky at first, but breaking it down simplifies the process. Let's graph the given piecewise function step-by-step.
For the function:
\[ y= \left\{ \begin{aligned} -2 & \text{ if } x \leq 3 \ 2 & \text{ if } x > 3 \end{aligned}\right. \]
We proceed as follows:
1. Start with the first piece, \( y = -2 \), for all \( x \leq 3 \). Draw a horizontal line at \( y = -2 \) extending from \( x = -fty \) to \( x = 3 \).
2. Place a filled circle at point (3, -2) to indicate that this point is included in the first part.
3. Next, draw the second piece, \( y = 2 \), for all \( x > 3 \). Again, draw a horizontal line at \( y = 2 \) starting from just above \( x = 3 \) continuing indefinitely toward \( x = fty \).
4. Place an open circle at point (3, 2) to indicate the function does not include this point but starts immediately after.
And there you have it! By following these steps, the graph is complete. It clearly shows the two separate pieces of the function and how they relate to the given x-values.