Problem 55 - Calculus with Concepts in Calculus

Question

Graph each pair of functions. Shade the region(s) the graphs enclose. $$ \begin{aligned} &g(x)=2 x / \sqrt{1+x^{2}} \text { and } k(x)=x / \sqrt{1-x^{2}} \text { (Hint: The }\\\ &\text { region has two parts.) } \end{aligned} $$

Step-by-Step Solution

Answer

Graph \( g(x) \) and \( k(x) \), and shade the regions where they overlap near the origin and symmetrically on the negative side.

Step 1: Understand the Functions

The given functions are \( g(x) = \frac{2x}{\sqrt{1+x^2}} \) and \( k(x) = \frac{x}{\sqrt{1-x^2}} \). These functions involve square roots in their denominators, indicating there are restrictions on their domains.

Step 2: Determine the Domain

For \( g(x) \), the domain is all real numbers because \( \sqrt{1+x^2} \) is defined for all \( x \). For \( k(x) \), the domain is \( -1 < x < 1 \) because \( \sqrt{1-x^2} \) is defined only when \( x^2 < 1 \).

Step 3: Analyze the Functions' Intersections

Set \( g(x) = k(x) \) to find intersections. This gives: \[ \frac{2x}{\sqrt{1+x^2}} = \frac{x}{\sqrt{1-x^2}} \] Simplifying, we get: \[ 2 \sqrt{1-x^2} = \sqrt{1+x^2} \] Squaring both sides and simplifying results in a complex expression that indicates analytical solutions might be difficult to find for actual intersection points.

Step 4: Graph the Functions

Graph \( g(x) \) and \( k(x) \) on a common coordinate plane. Use technology or graphing tools to accurately represent the functions and visualize their regions of intersection.

Step 5: Identify Regions Enclosed

The region enclosed by the graphs of \( g(x) \) and \( k(x) \) consists of two parts: one near the origin within the first intersection visible, and another symmetric near the negative side.

Step 6: Shade the Enclosed Regions

After identifying the regions enclosed by the graphs, shade these regions on the graph to highlight where \( g(x) \) and \( k(x) \) overlap. Each shaded region will illustrate where the functions enclose areas between them.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are essentially discussing the set of all possible input values (x-values) that will output a real number for the function. In simple terms, it is the range of x-values that you are allowed to use in the function without breaking any mathematical rules.
Understanding the domain is crucial because it tells us where the function is valid and thus, where we can graph it.

• For the function \( g(x) = \frac{2x}{\sqrt{1+x^2}} \), the expression \( \sqrt{1+x^2} \) in the denominator is defined for all real numbers. This is because no matter what value \( x \) takes, \( 1 + x^2 \) is always positive and greater than zero, so \( g(x) \) is defined everywhere on the real number line.
• On the other hand, \( k(x) = \frac{x}{\sqrt{1-x^2}} \) has a more restricted domain. The square root \( \sqrt{1-x^2} \) is only defined when \( 1-x^2 > 0 \), which means \( x^2 < 1 \). Thus, \( x \) must be between -1 and 1, so the domain of \( k(x) \) is \(-1 < x < 1\).

Understanding these domains provides boundaries for our graphs and informs the intervals where function interactions occur.

Function Intersection

Function intersection refers to the points where two functions have the same value; this means their graphs cross at these points. Finding intersections can give us critical information about how functions behave relative to each other.
To find intersections, we typically set the functions equal to each other and solve for x.

• For our functions \( g(x) \) and \( k(x) \), the equation \( \frac{2x}{\sqrt{1+x^2}} = \frac{x}{\sqrt{1-x^2}} \) suggests this point. Solving this can be complex, as you need to handle square roots and algebraic manipulation.
• After cross-multiplying and simplifying, these kinds of equations often lead to further algebraic expressions or require numerical methods to pinpoint exact intersection points.

Graphical methods or software tools can be very helpful at this step since they allow for a visual representation, which sometimes reveals intersections more clearly than pure algebraic attempts.

Graph Shading

Graph shading is an essential skill when dealing with functions and their graphs, especially when identifying the regions they enclose. After plotting the functions on a coordinate plane and determining their intersections, the next step is often to shade the areas between the graphs that satisfy a particular condition.
Shading helps to clarify the parts of the plane where a certain inequality holds true and visually delineates the regions of interest.

• For the functions \( g(x) \) and \( k(x) \), the solution suggests that the enclosed region is split into two parts, which can often be on either side of an intersection point.
• Once you've plotted both functions and identified these intersections using a graphing tool, use shading to indicate clearly where these functions overlap each other. Typically, this is where one function is above another over a certain domain.

Skillful graph shading enables you to highlight areas where certain conditions or relations between functions hold, making it easier to analyze and interpret graphically what's going on in the function plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph functions, using the horizontal axis as the x-axis and the vertical axis as the y-axis. This plane allows us to visually represent relationships between two variables, often given as functions or equations.

• Each point on the coordinate plane corresponds to an ordered pair \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position. This helps us sketch functions like \( g(x) \) and \( k(x) \).
• When graphing the functions, the x-values are taken from the domain of the function, and the corresponding y-values are calculated by substituting x into the function's equation.
• The coordinate plane is not just about plotting points; it's a powerful diagrammatic tool that supports the visualization of various mathematical concepts like function intersections, graph shading, and more.

The ability to represent functions on the coordinate plane effectively aids in understanding the structure and interaction of mathematical equations, providing insights into real-world applications and more complex mathematical analysis.