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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.40a
Chapter 7, Problem 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?

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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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Key Concepts

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

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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Linearization

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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Related Practice
Textbook Question

Find the inverse of the function f(x)=mx, where m is a constant different from zero.

Textbook Question

In Exercises 41–44:

a. Find f⁻¹(x).


43. f(x) = 5 − 4x, a = 1/2

Textbook Question

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)

Textbook Question

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

1
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Textbook Question

a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.

Textbook Question

88. Given that x>0, find the maximum value, if any, of

a. x^(1/x)