Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.1.71

Chapter 1, Problem 1.1.71

[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

Verified step by step guidance

Step 1: Begin by graphing the functions f(x) = x/2 and g(x) = 1 + (4/x). Use graphing technology or software to plot these functions on the same set of axes. Observe where the graph of f(x) is above the graph of g(x).

Step 2: To confirm the graphical findings algebraically, set up the inequality x/2 > 1 + 4/x. This inequality represents the condition where the graph of f(x) is above the graph of g(x).

Step 3: Clear the fraction by multiplying both sides of the inequality by x (assuming x ≠ 0) to obtain x^2/2 > x + 4.

Step 4: Rearrange the inequality to form a quadratic inequality: x^2/2 - x - 4 > 0. Multiply through by 2 to eliminate the fraction, resulting in x^2 - 2x - 8 > 0.

Step 5: Solve the quadratic inequality x^2 - 2x - 8 > 0. Factor the quadratic expression or use the quadratic formula to find the critical points. Determine the intervals where the inequality holds true by testing values within the intervals defined by the critical points.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

10m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Functions

Graphing functions involves plotting them on a coordinate plane to visually analyze their behavior and intersections. For f(x) = x/2 and g(x) = 1 + (4/x), graphing helps identify where one function is greater than the other by observing their curves and intersection points. This visual representation is crucial for understanding inequalities and relationships between functions.

Inequalities

Inequalities express a relationship where one expression is greater or less than another. In this context, x/2 > 1 + 4/x requires determining the values of x for which the function f(x) is greater than g(x). Solving inequalities often involves algebraic manipulation and understanding the behavior of functions, especially when they involve rational expressions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities. To confirm graphically identified solutions algebraically, one must manipulate the inequality x/2 > 1 + 4/x, potentially by finding a common denominator or isolating terms. This process is essential for verifying solutions and understanding the underlying mathematical relationships.

Recommended video:

05:25

Determine Continuity Algebraically

Related Practice

Textbook Question

Algebraic Combinations

In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.

f(x) = √(x + 1), g(x) = √(x − 1)

241

views

Textbook Question

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = (−x)²/³

188

views

Textbook Question

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = x + (1/10) sin 30x

184

views

Textbook Question

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

sin² 3π/8

304

views

Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = e⁻ˣ²