Domanda del libro di testo
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)
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 1, Problema 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.
Guida verificata passo dopo passo1
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.
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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.
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function
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.
Video consigliato:
Percorso guidato
5:10Finding the Domain and Range of a Graph
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.
Video consigliato:
05:25Determine Continuity Algebraically
Pratica correlata
Domanda del libro di testo
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
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Domanda del libro di testo
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
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Domanda del libro di testo
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
sin² 3π/8
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Domanda del libro di testo
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
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Domanda del libro di testo
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
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