Problem 51 - Contemporary Precalculus

Question

Sketch the graph of a function \(f\) that satisfies these five conditions: (i) \(f(-1)=2\) (ii) \(f(x) \geq 2\) when \(x\) is in the interval \(\left(-1, \frac{1}{2}\right)\) (iii) \(f(x)\) starts decreasing when \(x=1\) (iv) \(f(3)=3=f(0)\) (v) \(f(x)\) starts increasing when \(x=5\) [Note: The function whose graph you sketch need not be given by an algebraic formula.]

Step-by-Step Solution

Answer

In summary, to sketch the graph of the function \(f(x)\) that meets the five conditions, we plotted the given points, applied the inequality condition, and considered the decreasing and increasing intervals. By combining all the information, we were able to create a graph with a horizontal line at \(y=2\) between \((-1, 2)\) and \((\frac{1}{2}, 2)\), a connection from (0,3) to \((\frac{1}{2}, 2)\), a decreasing portion when \(x=1\), a connection from (1,1) to (3,3), and an increasing portion when \(x=5\). This graph fulfills all the given conditions.

Step 1: Plot the given points

First, we should plot the points we know: \((-1, 2)\) and \((3, 3)\). Additionally, we know there is a point on the graph with \(x=0\), and it has a value \(f(0) = 3\).

Step 2: Apply the inequality condition

We are given that \(f(x) \geq 2\) when \(x\) is in the interval \(\left(-1, \frac{1}{2}\right)\). This means that for this interval, the function should stay above or on the line \(y=2\).

Step 3: Consider decreasing/increasing intervals

We know that \(f(x)\) starts decreasing when \(x=1\), so from this point, the function will begin to decrease. At \(x=5\), the function begins to increase again.

Step 4: Combining the information to create a graph

Based on the conditions, 1. To satisfy condition (ii), draw a horizontal line at \(y=2\) from \((-1,2)\) until $x=\frac{1}{2}. 2. Connect the point \((0,3)\) and \((\frac{1}{2},2)\) and make sure the graph stays above the line \(y=2\) in this region. 3. Start decreasing the graph when \(x=1\) to fulfill the condition (iii). 4. Connect the point \((1,1)\) to the point \((3,3)\). 5. Lastly, begin increasing the function when \(x=5\) and continue to the right to satisfy the condition (v). The completed graph should show a function that meets all five conditions and represents the behavior described in the problem.

Key Concepts

Graph Sketching

Graph sketching is a fundamental skill in precalculus that helps you visualize the behavior of a function. By sketching a graph, you can better understand various function characteristics such as key points, and intervals where the function changes its behavior. When sketching graphs, you should:

• Identify and plot known points on the graph. For example, the points \((-1, 2)\), \((3, 3)\), and \((0, 3)\) are essential markers for our graph.
• Interpret conditions given for the function. In the exercise, these include function values, increasing or decreasing tendencies, and inequalities.
• Connect the dots considering the overall behavior such as increasing, decreasing, or holding a steady value over intervals.

Breaking the problem down in this way can make graph sketching less intimidating and more intuitive.

Function Behavior

Understanding function behavior involves comprehending how the function changes over its domain. Functions are dynamic, and recognizing key behaviors helps in sketching and analyzing graphs. In our case, the function exhibits distinct characteristics:

• The function holds steady at certain values, such as where \(f(x)\geq 2\) in the interval \((-1, \frac{1}{2})\).
• There are specific points where the function's tendency changes. For instance, \(f(x)\) decreases after \(x=1\) and increases again past \(x=5\).

These behaviors are crucial for creating an accurate graph where each section of the function reflects the given conditions. By capturing these behaviors, you grasp a comprehensive view of what the function does at any given region of its domain.

Inequalities

Inequalities arise often in function analysis, providing constraints or conditions that dictate where the function lies relative to certain values or lines. In this context, the inequality \(f(x) \geq 2\) within \(x \in (-1, \frac{1}{2})\) suggests that the graph must lie on or above the line \(y=2\) in this interval.

• Draw a horizontal line at \(y=2\) to represent the boundary. The function must remain on or above this line between \(-1\) and \(\frac{1}{2}\).
• Ensure any connecting lines between plotted points also respect this condition, by residing above or on the line within the given interval.

Handling inequalities properly is critical in graph sketching because it strictly defines parts of the function's range and ensures the graph's accuracy in reflecting these constraints.

Increasing and Decreasing Functions

Identifying when a function is increasing or decreasing is vital for understanding its overall shape. This involves recognizing intervals where the slope of the function is positive (increasing) or negative (decreasing). For our exercise:

• Beginning at \(x=1\), the function decreases, which you can illustrate by a downward slope.
• Once reaching \(x=5\), the function begins to increase again. This action is depicted through an upward sloping segment.

This involves drawing line segments or curves that accurately reflect these behaviors. By linking these tendencies to key points, you can depict how the function naturally curves or levels, establishing its overall graphical structure.