Problem 53 - A Graphical Approach to Precalculus with Limits

Question

In 2017 , the price in dollars for first-class letters weighing up to 5 ounces could be computed by the piece wise constant function \(f,\) where \(x\) is the number of ounces. $$f(x)=\left\\{\begin{array}{ll} 0.49 & \text { if } 0< x \leq 1 \\ 0.70 & \text { if } 1< x \leq 2 \\ 0.91 & \text { if } 2< x \leq 3 \\ 1.12 & \text { if } 3< x \leq 4 \\ 1.33 & \text { if } 4< x \leq 5 \end{array}\right.$$ (a) Evaluate \(f(1.5)\) and \(f(3) .\) Interpret your results. (b) Sketch a graph of \(f .\) What is its domain? What is its range?

Step-by-Step Solution

Answer

Evaluations: \(f(1.5)=0.70\), \(f(3)=0.91\). Domain: \((0, 5]\); Range: \(\{0.49, 0.70, 0.91, 1.12, 1.33\}\).

Step 1: Evaluate f(1.5)

To evaluate \( f(1.5) \), observe that \( 1.5 \) falls within the interval \( 1 < x \leq 2 \). According to the piecewise function definition, for this interval, \( f(x) = 0.70 \). Therefore, \( f(1.5) = 0.70 \).

Step 2: Evaluate f(3)

For \( f(3) \), notice that \( 3 \) is on the boundary of the interval \( 2 < x \leq 3 \). According to the piecewise function, \( f(x) = 0.91 \) in this interval, thus \( f(3) = 0.91 \).

Step 3: Interpret the Results

The value \( f(1.5) = 0.70 \) means that a letter weighing 1.5 ounces costs \(0.70. Similarly, \( f(3) = 0.91 \) indicates that a letter weighing 3 ounces costs \)0.91.

Step 4: Sketch the Graph

To sketch the graph, draw horizontal line segments corresponding to each interval: from \( x = 0 \) to \( x = 1 \) at \( y = 0.49 \), from \( x = 1 \) to \( x = 2 \) at \( y = 0.70 \), from \( x = 2 \) to \( x = 3 \) at \( y = 0.91 \), from \( x = 3 \) to \( x = 4 \) at \( y = 1.12 \), and from \( x = 4 \) to \( x = 5 \) at \( y = 1.33 \).

Step 5: Determine the Domain

The domain of \( f \) is the set of all \( x \) values for which \( f(x) \) is defined. Here, \( x \) varies from just above 0 to 5 (since \( 0 < x \leq 5 \)). Thus, the domain is \( (0, 5] \).

Step 6: Determine the Range

The range of \( f \) is the set of all possible output values for \( f(x) \). The values are \( 0.49, 0.70, 0.91, 1.12, 1.33 \) as each interval is constant. Therefore, the range is \( \{0.49, 0.70, 0.91, 1.12, 1.33\} \).

Key Concepts

Domain and Range

The **domain** and **range** are critical concepts when working with piecewise functions. The **domain** of a function represents all the input values (or x-values) over which the function is defined. In this piecewise function that determines the price of mailing letters by weight, the domain is from 0 to 5 ounces, exclusive of 0 and inclusive of 5. This is denoted as \((0, 5]\).

Essentially, any weight within this interval has a defined price. The **range** of a function includes all possible output values, or y-values, which a function can produce based on its domain. For our mailing price function, each weight interval corresponds to a constant price. Thus, the range is simply the set of different prices that appear across the intervals: \( \{0.49, 0.70, 0.91, 1.12, 1.33\} \).

This collection of prices highlights how the cost changes based on weight, making the function handy for price estimation within its domain.

Graphing Functions

Graphing a piecewise function involves plotting each part of the function on the coordinate grid. Each segment is determined by its constant value over a specified interval. For the piecewise function for mailing costs:

• From \( x = 0\) to \( x = 1\), the graph is a horizontal line at \( y = 0.49\).
• From \( x = 1\) to \( x = 2\), the line jumps to \( y = 0.70\).
• From \( x = 2\) to \( x = 3\), it continues at \( y = 0.91\).
• From \( x = 3\) to \( x = 4\), the line rises to \( y = 1.12\).
• Finally, from \( x = 4\) to \( x = 5\), it reaches \( y = 1.33\).

Each of these pieces is a continuous horizontal line segment, reflecting a constant price over its interval. Take note of the boundaries: the endpoint where \( x \) reaches the next weight limit will determine whether the dot is filled (inclusive) or open (exclusive), demonstrating whether the boundary value is part of the interval.

Function Evaluation

To evaluate a piecewise function, you need to determine which part of the function the particular x-value falls into. In our case, if you want to find the price for mailing a letter weighing 1.5 ounces, locate which interval 1.5 falls into. Since it’s within \( 1 < x \leq 2 \), the function value \( f(1.5) = 0.70\).

Similarly, for a letter weighing exactly 3 ounces, we locate that 3 fits within \( 2 < x \leq 3\), giving \( f(3) = 0.91\). Function evaluation in piecewise functions is straightforward: Ensure the x-value matches the correct interval, then apply the constant function value defined for that interval. This straightforward process allows you to quickly find function values and interpret real-life costs, as in this mailing cost example.