Problem 31 - Thomas Calculus

Question

a. Graph the functions \(f(x)=x / 2\) and \(g(x)=1+(4 / x)\) together to identify the values of \(x\) for which $$ \frac{x}{2}>1+\frac{4}{x} $$ b. Confirm your findings in part (a) algebraically.

Step-by-Step Solution

Answer

Graphically and algebraically, the inequality \( \frac{x}{2} > 1 + \frac{4}{x} \) holds for \( x > 4 \) and \( x < -2 \).

Step 1: Understand the Functions

We have two functions to graph: \( f(x) = \frac{x}{2} \) and \( g(x) = 1 + \frac{4}{x} \). The function \( f(x) \) is a linear function with a slope of \( \frac{1}{2} \), while \( g(x) \) is a rational function that varies depending on \( x \). Our goal is to find the range of \( x \) where \( \frac{x}{2} > 1 + \frac{4}{x} \).

Step 2: Graph the Functions

Plot \( f(x) = \frac{x}{2} \), which is a straight line through the origin with a slope of \( \frac{1}{2} \). Then plot \( g(x) = 1 + \frac{4}{x} \), starting with the undefined point at \( x = 0 \), and notice that \( g(x) \) decreases as \( x \) increases, but increases again for negative \( x \). Identify where \( f(x) \) is above \( g(x) \).

Step 3: Identify the Intersection Points

Find the points of intersection by setting \( \frac{x}{2} = 1 + \frac{4}{x} \). Solve the equation \( \frac{x}{2} = 1 + \frac{4}{x} \) to find potential boundary points by multiplying both sides by \( x \) and arranging terms.

Step 4: Solve the Equation

Solving \( \frac{x}{2} = 1 + \frac{4}{x} \) gives:\[ \frac{x^2}{2} = x + 4 \]\[ x^2 - 2x - 8 = 0 \]Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots of this equation.

Step 5: Use the Quadratic Formula

Apply the quadratic formula with \( a = 1, b = -2, \) and \( c = -8 \):\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \]\[ x = \frac{2 \pm \sqrt{4 + 32}}{2} \]\[ x = \frac{2 \pm \sqrt{36}}{2} \]\[ x = \frac{2 \pm 6}{2} \]This yields solutions: \( x = 4 \) and \( x = -2 \).

Step 6: Determine the Inequality Solution

Between the critical points \( x = -2 \) and \( x = 4 \), graphically, \( \frac{x}{2} > 1 + \frac{4}{x} \) holds for \( x > 4 \) and \( x < -2 \). These intervals are the solution to the inequality.

Key Concepts

Linear Functions

A linear function is a fundamental concept in algebra and functions, represented in the form \( f(x) = mx + b \). This expression outlines a straight line when plotted on a Cartesian coordinate system. Here, \( m \) embodies the slope of the line, indicating how steep the line is. Meanwhile, \( b \) serves as the y-intercept, which is where the line crosses the y-axis.
\( f(x) = \frac{x}{2} \) offers a great example of a linear function with a slope \( m = \frac{1}{2} \), meaning the line rises half a unit for every one unit it moves horizontally. This specific line passes through the origin point \((0, 0)\), due to \( b \) equating to zero. A keenness to interpret these aspects of linear functions enhances one's grasp on linear equations and inequalities.
Linear functions are often involved in forming linear equations or inequalities, just like our exercise. Understanding their graphical representation offers a visual way to solve and comprehend equations, making graphing an indispensable skill.

Rational Functions

Rational functions are quotients of two polynomials, represented as \( g(x) = \frac{p(x)}{q(x)} \). These functions showcase distinct behaviors based on the variable's value, leading to many interesting properties, such as undefined points and asymptotes.
The function \( g(x) = 1 + \frac{4}{x} \), can be split into a constant and a rational part. It includes a vertical asymptote at \( x = 0 \), where the function is undefined. For positive \( x \) values, as \( x \) increases, the term \( \frac{4}{x} \) decreases, causing \( g(x) \) to approach 1. Conversely, for negative \( x \), it decreases significantly and changes behavior.
The graph of rational functions may not resemble a straight line like linear functions, as they approach certain values infinitely but never actually reach them. Understanding and visualizing these characteristics mark a crucial step in mastering function analysis and solving rational inequalities.

Quadratic Equations

Quadratic equations take the general form \( ax^2 + bx + c = 0 \). They represent a parabola when graphed, with the vertex being the curve's peak or trough depending on the leading coefficient. Solving these equations involves finding where the parabola intersects the x-axis, known as solving for roots or zeros.
In our original exercise solution, we derived a quadratic equation \( x^2 - 2x - 8 = 0 \) from setting the linear function equal to the rational one. This quadratic equation employed the Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find its roots \( x = 4 \) and \( x = -2 \). These results indicate the critical points for evaluating the inequality.
By identifying roots and understanding a parabola's nature, solving quadratic equations helps greatly in tackling inequalities and examining intersections between different functions.