Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}-1 & \text { if } x \text { is an integer } \\ -2 & \text { if } x \text { is not an integer }\end{cases} $$

Step-by-Step Solution

Answer

The graph has dots at \(y = -1\) for integer \(x\) and a line at \(y = -2\) for non-integer \(x\).

Step 1: Understanding the Function Definition

The function \( f(x) \) is defined as a piecewise function. It takes two different values: \(-1\) when \(x\) is an integer and \(-2\) when \(x\) is not an integer.

Step 2: Identifying Integer Values

List or note all integer values of \(x\) within the domain you're considering. For a more general graph, integers like -3, -2, -1, 0, 1, 2, 3, etc., should be considered.

Step 3: Plotting Values for Integer Inputs

For each integer \(x\), plot a point at \((x, -1)\). These points will form a horizontal line at \(y = -1\) but only where \(x\) is an integer. This represents the first case in our piecewise function.

Step 4: Plotting Values for Non-Integer Inputs

For all non-integer values of \(x\), plot a point at \((x, -2)\). This forms a continuous horizontal line at \(y = -2\) for all non-integer \(x\). This represents the second case of the piecewise function.

Step 5: Finalizing and Sketching the Graph

Combine the plotted points into a coherent graph. The graph consists of separate horizontal point-lined segments at \(y = -1\) for integer \(x\) and a solid horizontal line at \(y = -2\) for non-integer \(x\). Make sure the points for \(y = -1\) are clear and distinct on this graph.

Key Concepts

Graphing

A piecewise function can be visualized using graphing, which helps us easily see the behavior of the function over different sections of its domain. When graphing a piecewise function like \(f(x)\), we first identify the different cases provided in the function's definition. Each case will correspond to a specific line or point on the graph. In our case, for \(f(x)\), we have two different horizontal lines:

• A horizontal line at \(y = -1\), corresponding to integer values of \(x\).
• A horizontal line at \(y = -2\), for non-integer values of \(x\).

Graphing this involves plotting each of these lines separately and ensuring they follow the rules set out by the piecewise definition. This method helps in breaking down the graph into understandable parts, making it easier to see where each case applies on the graph.

Integer Values

Integer values are specific values of \(x\) where the function \(f(x) = -1\). Each integer leads to one plot point on the graph located at \((x, -1)\). Common integers like \(-3, -2, -1, 0, 1, 2, 3\) are easy examples. When we plot these points, they collectively appear as a dotted horizontal line along \(y = -1\).

• Identify integers within the chosen domain.
• Plot exactly at these integers, avoiding any fractional or decimal numbers.
• Use distinct dots or circles to indicate these discrete points.

By focusing on integer values separately, you clearly demonstrate one part of the piecewise function, making it easier to distinguish from the continuous elements when graphing other parts of the function.

Non-Integer Values

Non-integer values, such as fractions and decimals, correspond to the second case in the function \(f(x) = -2\). For any value of \(x\) that is not an integer, the function remains constant at \(y = -2\).
This results in a solid horizontal line at \(y = -2\), representing all non-integer inputs such as \(x = 0.5, x = 1.8\), etc.

• Focus on values that aren't whole numbers, even if they are complex like square roots or irrational numbers.
• The line formed by these points is seamless and continuous, clearly distinguishing it from the dotted one for integer values.
• Ensuring non-integers cover the gaps left by the integers helps complete the visual representation of the piecewise function.

By paying attention to non-integer inputs, we get a complete view of the function across all possible values of \(x\), showcasing how a piecewise function is composed of distinct but perfectly complementing parts.