Problem 32 - Applied Calculus

Question

Use a graphing calculator to graph each piecewise nonlinear function on the window [-2,10] by [-5,5] . Where parts of the graph do not touch, state which point is included and which is excluded. $$ f(x)=\left\\{\begin{array}{ll} 4-x^{2} & \text { if } x < 3 \\ 2 x-11 & \text { if } 3 \leq x < 7 \\ 8-x & \text { if } x \geq 7 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph each piece separately on the graphing calculator: ensure \((3, -5)\) is included and \((7, 3)\) is excluded.

Step 1: Understand the Function

The function is piecewise and has three parts, each applicable to different intervals of \(x\). The first part \(4-x^2\) is valid for \(x < 3\). The second part \(2x-11\) is for \(3 \leq x < 7\). The third part \(8-x\) applies for \(x \geq 7\). Each part of the function will be graphed separately according to these conditions.

Step 2: Graph the First Piece

For the interval \(x < 3\), graph \(4-x^2\). This is a downward opening parabola with vertex at the origin, which continues indefinitely to the left. When \(x = 3\), the function evaluates to \(-5\), but this point is not included in the graph (open circle) because \(x < 3\).

Step 3: Graph the Second Piece

Next, graph \(2x-11\) on the interval \(3 \leq x < 7\). This is a linear function, a straight line, starting from the point \((3, -5)\) (closed circle, point included) to the point \((7, 3)\) (open circle, point excluded).

Step 4: Graph the Third Piece

Graph \(8-x\) for \(x \geq 7\). This is another linear function starting from the point \((7, 1)\) (closed circle, point included) and decreasing indefinitely. Ensure each segment is accurately represented on the window \([-2, 10] \) and \([-5, 5]\).

Step 5: Verify Graph on the Calculator

Input each piece of the function in the graphing calculator, restricting the domain for each part appropriately to visualize them all on the same set of axes. Ensure the correct window settings and points are displayed with their respective inclusions and exclusions.

Key Concepts

Using a Graphing Calculator Effectively

A graphing calculator is an invaluable tool for visualizing piecewise nonlinear functions. This technological marvel simplifies complex algebraic graphs, making them accessible and understandable. To start using a graphing calculator for piecewise functions:

• Begin by entering each piece of the function separately. Ensure that each segment is defined on its specific interval.
• The window settings should be adjusted to view the relevant portion of the graph. For this task, set the window from [-2, 10] for the x-axis and [-5, 5] for the y-axis.
• Check if each interval is properly marked, meaning the open and closed circles representing included and excluded points are clearly visible.

This approach ensures every part of the piecewise function is properly displayed. It helps in understanding the transitions and breaks between each segment.

Understanding Nonlinear Functions

Nonlinear functions are those where the relationship between the variables doesn't form a straight line when graphed. Instead, they may form curves or other shapes. For the given piecewise function, we're dealing with both nonlinear and linear functions.

• The first piece, \(4-x^2\), is a nonlinear function. It is a parabola that opens downwards.
• The second piece, \(2x-11\), is a linear function, but as part of a piecewise definition, it involves nonlinear transitions at boundaries.
• The third piece, \(8-x\), is linear, creating additional transitions between different function behaviors when combined.

Each function type has distinct properties in a graph. Understanding them individually aids comprehension of the whole piecewise function.

Introduction to Interval Notation

Interval notation is a crucial mathematical concept that clarifies the range of values a function or part of a function operates over. In piecewise functions, interval notation helps define where each function piece is applicable.

• For interval \(x < 3\), the notation \((-\infty, 3)\) is used. This includes all numbers less than 3 but excludes 3 itself, indicated by the use of a parenthesis.
• In \(3 \leq x < 7\), the interval notation is \([3, 7)\), where the square bracket means 3 is included.
• For \(x \geq 7\), the notation \([7, \infty)\) signifies that the range includes 7 and extends indefinitely.

Grasping interval notation is key to understanding how to apply and interpret each piece of a piecewise function.