Problem 8 - Calculus: An Intuitive and Physical Approach

Question

Graph the following functions: (a) \(y=x+\sqrt{(1-x)(2-x)}\) (b) \(y=x+\sqrt{(x-2)(x+1)}\)

Step-by-Step Solution

Answer

Graph of (a) includes points on the interval \(x \: \leq 1\); graph of (b) includes points on the interval \(x \: \geq 2\). Both graphs exhibit curves due to the square root component.

Step 1 - Determine the domain of (a)

The function under the square root must be non-negative. For the function (a) \(y = x + \sqrt{(1-x)(2-x)}\), this means \(1-x\geq 0\) and \(2-x \: >0\). Solve these inequalities to find the domain.

Step 2 - Solve inequalities for (a)

Solving \(1-x\geq 0\) gives \(x \: \leq 1\), and \(2-x > 0\) gives \(x < 2\). Therefore, the domain of the function (a) is \(x \: \leq 1\).

Step 3 - Plot the function (a)

Using the domain, for every value of \(x\) within the domain, calculate the corresponding value of \(y\) and plot the point \( (x, y) \) on the graph. Connect these points smoothly to reflect the function (a), considering that the square root function produces a curve part.

Step 4 - Determine the domain of (b)

For the function (b) \(y = x + \sqrt{(x-2)(x+1)}\), we require \(x-2 \: \geq 0\) and \(x+1 \: \geq 0\). Solve these inequalities to find the domain.

Step 5 - Solve inequalities for (b)

Solving \(x-2 \: \geq 0\) gives \(x \: \geq 2\), and \(x+1 \geq 0\) gives \(x \: \geq -1\). Thus, the domain of function (b) is \(x \: \geq 2\).

Step 6 - Plot the function (b)

Using the domain, calculate the value of \(y\) for each value of \(x\) within the domain and plot the point \( (x, y) \) on the graph. Connect these points carefully to illustrate the function (b), taking into account the impact of the square root function.

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is crucial when working with various mathematical functions, especially when determining their domain. Inequalities tell us how numbers compare with each other, indicating whether one number is less than, greater than, less than or equal to, or greater than or equal to another number.

Solving an inequality is similar to solving a regular equation, but with an important difference: when you multiply or divide both sides of the inequality by a negative number, you must flip the inequality sign. For example, if we have the inequality \( -2x > 6 \), and we divide both sides by -2, we must reverse the > sign to <, giving us \( x < -3 \).

• Always perform the same operation on both sides of the inequality.
• Pay attention when dealing with negative numbers.
• Remember to flip the inequality sign when needed.

By mastering solving inequalities, one can confidently determine function domains and set proper constraints for graphing.

Function Domain

The domain of a function is the complete set of possible values of the independent variable, often denoted as \(x\), over which the function is defined. It is important to determine the domain to understand where the function exists and to avoid undefined or nonreal numbers in calculations.

When working with square root functions, as in our example \( y = x + \sqrt{(1-x)(2-x)} \), the domain is found by setting the expression under the square root to be greater or equal to zero since the square root of a negative number is not a real number. Solving the inequalities that arise from this condition, as shown in the steps, will provide you with the valid range of \(x\) values.

Square Root Functions

Square root functions are types of radical functions where the independent variable \(x\) is under the square root. The most basic form of a square root function is \(y=\sqrt{x}\), which produces a curve that starts at the origin (0,0) and extends into the first quadrant of the coordinate plane for positive \(x\).

With square root functions like those in the exercise, it's key to remember that the output \(y\) will always be a non-negative value because you cannot take the square root of a negative number and get a real result. That’s why understanding the domain of these functions protect us from ending up with undefined or nonreal values when plotting them.

Plotting Points

Plotting points is the process of drawing individual coordinates on a graph based on the \(x\) and \(y\) values that satisfy the function. For continuous functions, individual points are connected to show the shape of the graph. When dealing with functions that include the square root, we often see a curve that mirrors the basic shape of \(y=\sqrt{x}\), though it may be shifted or reflected depending on the specific function.

To plot points, determine the \(x\) value within the domain and then calculate the corresponding \(y\) value by plugging \(x\) back into the function. Repeat this process for multiple points, and then draw a smooth line or curve through those points to complete the graph. Having an accurate graph allows for better visualization and understanding of a function's behavior.