Problem 27 - Brief Applied Calculus

Question

$$ \text { Graph each function } $$ $$ f(x)=\left\\{\begin{array}{ll} 2 x-7 & \text { if } x \geq 4 \\ 2-x & \text { if } x<4 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the two lines for different intervals, ensuring correct domain restrictions.

Step 1: Understand the Piecewise Function

The given function is a piecewise function, which means it has different expressions for different intervals of the domain. Here, the function has two parts: \( f(x) = 2x - 7 \) when \( x \geq 4 \) and \( f(x) = 2 - x \) when \( x < 4 \).

Step 2: Identify the Domain for Each Piece

For the first piece \( 2x - 7 \), the domain is \( x \geq 4 \). For the second piece \( 2 - x \), the domain is \( x < 4 \). Understanding the domain helps in plotting the pieces correctly on the graph.

Step 3: Plot the First Piece

For \( x \geq 4 \), the function is \( f(x) = 2x - 7 \). Calculate a few points such as \( x = 4 \), \( x = 5 \): \( f(4) = 2(4) - 7 = 1 \); \( f(5) = 2(5) - 7 = 3 \). Draw a line through these points, starting at \( x = 4 \) and extending to the right since \( x \) is greater than or equal to 4.

Step 4: Plot the Second Piece

For \( x < 4 \), the function is \( f(x) = 2 - x \). Calculate a few points such as \( x = 3 \), \( x = 2 \): \( f(3) = 2 - 3 = -1 \); \( f(2) = 2 - 2 = 0 \). Draw a line through these points, stopping just before \( x = 4 \) to ensure the function does not include \( x = 4 \) in this interval.

Step 5: Combine the Pieces on the Graph

On the graph, combine both pieces ensuring continuity in the correct domains: the line from Step 3 starts at \( x = 4 \) and extends to the right; the line from Step 4 extends to the left from just below \( x = 4 \). Check for any gaps or undefined points especially at the boundary \( x = 4 \).

Key Concepts

graphing piecewise functions

Graphing piecewise functions involves plotting different equations within specific intervals on the same coordinate plane. These functions can be quite flexible, allowing different behaviors in different domains.
Start by analyzing the given piecewise function and identifying each distinct piece. Each piece will pertain to a section of the function's domain, often split by specific values of the variable, such as the condition of being less than or greater than a number. For example, in the function given,

• The piece \( 2x - 7 \) applies when \( x \geq 4 \).
• The piece \( 2 - x \) applies when \( x < 4 \).

After identifying these pieces, calculate a few key points for each part to aid in constructing each line. Make sure you respect each interval's boundaries, ensuring accurate representation of where each part of the function applies. Once you have your points plotted, connect them as lines or curves that are required by each piece's equation. Then, ensure that these segments meet correctly at their interval boundaries.

domain of piecewise functions

The domain of piecewise functions refers to the full set of input values for which the function is defined. For each segment of a piecewise function, the domain is explicitly given or implied by inequalities or conditions.

In our example, understanding the individual intervals of each piece is crucial to plot the function correctly.

• The domain for \( 2x - 7 \) is \( x \geq 4 \). This means the function part will include points like \( x = 4, \ 5, \ 6,\) and extend infinitely toward positive infinity.
• The domain for \( 2 - x \) is \( x < 4 \), encompassing points such as \( x = 3,\ 2, \ 1,\) and so forth.

Correctly distinguishing each part's domain ensures that no values are missed or incorrectly assigned, which is essential for both the function's accuracy and your understanding of the graph.

plotting functions in intervals

Plotting functions in intervals requires organizing your graph into segments according to specific intervals dictated by the function's definition. This involves calculating specific points within each defined interval to guide accurate plotting.

For piecewise functions, each function piece has its interval, which acts like boundaries:

• For \( x \geq 4 \), calculate points such as \( f(4) = 1 \) and \( f(5) = 3 \). These points define a line segment starting at \( x = 4 \) and extending rightwards.
• For \( x < 4 \), compute points like \( f(3) = -1 \) and \( f(2) = 0 \). This creates a line segment extending leftwards from just below \( x = 4 \).

By paying careful attention to where intervals start and stop, such as marking an open circle on the graph where an interval does not include an endpoint, you ensure clarity and precision in your graph. This not only represents the functions correctly but also aids in visual comprehension of how each piece fits within the whole function.