Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
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Ch. 7 - Transcendental 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 7, Problema 7.1.40a
Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
Guida verificata passo dopo passo1
To find the inverse of the function \(f(x) = -x + 1\), start by replacing \(f(x)\) with \(y\): write \(y = -x + 1\).
Next, swap the variables \(x\) and \(y\) to find the inverse function: write \(x = -y + 1\).
Now, solve this equation for \(y\) to express the inverse function: rearrange to get \(y\) in terms of \(x\).
For graphing, plot the original line \(y = -x + 1\) and the line \(y = x\) on the same coordinate plane. These lines will intersect at a point which you can find by setting \(-x + 1 = x\) and solving for \(x\).
To find the angle of intersection between the two lines, use the formula for the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\): \(\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\). Here, identify the slopes of the lines and substitute them into this formula.
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Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For f(x) = -x + 1, the inverse is found by solving y = -x + 1 for x and then expressing y in terms of x. The inverse function reflects the original function across the line y = x.
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Inverse Cosine
Graphing Linear Functions
Graphing linear functions involves plotting lines based on their slope and y-intercept. The line y = -x + 1 has a slope of -1 and intercept 1, while y = x has slope 1 and intercept 0. Visualizing both lines helps understand their relationship and points of intersection.
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Percorso guidato
07:17Linearization
Angle Between Two Lines
The angle between two intersecting lines can be found using their slopes. If m1 and m2 are slopes, the angle θ satisfies tan(θ) = |(m2 - m1) / (1 + m1*m2)|. This formula calculates the acute angle where the lines cross, important for understanding their geometric relationship.
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Percorso guidato
05:13Slopes of Tangent Lines
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Domanda del libro di testo
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
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Domanda del libro di testo
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)
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89. Use limits to find horizontal asymptotes for each function.
a. y = x tan(1/x)
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Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. a. arccos(1/2)
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Domanda del libro di testo
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
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