Problem 25 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. The function \(f\) defined by $$ f(x)=\left\\{\begin{array}{ll} 3 & \text { for } x \neq 1 \\ 5 & \text { for } x=1 \end{array}\right. $$ A function whose graph resembles that of \(f\) can be called a "hiccup" function.

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = 3 \) with a closed circle at \( (1, 5) \).

Step 1: Analyze the Function Definition

The function \( f(x) \) is defined in two parts. For most values of \( x \), specifically when \( x eq 1 \), \( f(x) = 3 \). However, at the particular point \( x = 1 \), \( f(x) \) takes the value 5. This indicates that the graph will mostly be horizontal at \( y = 3 \) with an exception at \( x = 1 \).

Step 2: Plot the Points for \( x \neq 1 \)

Since \( f(x) = 3 \) for all \( x eq 1 \), the graph will be a horizontal line at \( y = 3 \). Draw this horizontal line across the graph except at the point \( (1, 3) \). Use an open circle to indicate that this value is not included at \( x = 1 \).

Step 3: Plot the Special Point \( x = 1 \)

For \( x = 1 \), \( f(x) = 5 \). On the graph, place a closed circle at the point \( (1, 5) \). This marks the exception where the function takes the value 5.

Step 4: Review the Graph

Check that the graph has a horizontal line at \( y = 3 \) with an open circle at \( (1, 3) \) and a closed circle at \( (1, 5) \). This creates a gap at \( x = 1 \), giving it the appearance of a small "hiccup."

Step 5: Draw Conclusion

The graph represents a function that is constant except for a sudden change at one point, hence resembling a "hiccup" at \( x = 1 \).

Key Concepts

Graphing Discontinuous Functions

Graphing discontinuous functions might sound tricky, but it's actually about recognizing where the function breaks or changes suddenly. In general, a function is discontinuous at points where it has an abrupt change in value, making the graph jump from one point to another.
Discontinuities can appear as "holes" in the graph or as a "jump," where the graph seems to suddenly leap to another value. This is exactly what we see in the hiccup function, where the value jumps from 3 to 5 at a specific point. Visualizing these gaps or jumps is crucial for understanding how discontinuous functions behave.
When graphing such functions, pay attention to:

• Points where the function is undefined or takes a different value (often marked by open or closed circles).
• The overall shape or pattern of the graph outside these unusual points.
• Changes in the direction or slope of the graph at the point of discontinuity.

A good practice is to always review your graph to ensure all discontinuities are clearly depicted.

Step Functions

Step functions are unique types of discontinuous functions that resemble staircases when graphed. They start one way, stay constant for a stretch, and then abruptly step up or down to a different value.
Think of them as a series of constant segments. Each "step" indicates a sudden change in value, and this creates a "step-like" appearance on the graph.
Characteristically, step functions have:

• Constant sections that are parallel to the x-axis (horizontal lines).
• Suden jumps at certain points called 'step points'. These usually occur at integer values of x.
• Distinct vertical breaks or jumps between different constant sections.

These functions are often used to model processes where changes occur in instant bursts rather than gradually, such as ticket prices or tax brackets that change suddenly once a threshold is passed.

Function Graph Sketching

Function graph sketching is the art of drawing the graphical representation of a mathematical function. This involves plotting points on a coordinate system and connecting them, following the specific patterns dictated by the function's rules.
When sketching a function, especially one with discontinuities, the following steps can be helpful:

• Identify key points, such as where the function has breaks or jumps.
• Understand the overall layout. Are there any repeating patterns, or is it mostly constant?
• Use circles (open for points not included, closed for included values) to indicate specific changes.
• Ensure any discontinuities or 'steps' are clearly marked on the graph.

Graph sketching is not just about plotting points but understanding how the function behaves, flows, and changes. It makes mathematical concepts tangible and visual.