Textbook Question
In Exercises 5–8, show that each function is a solution of the given initial value problem.
5. Differential Equation: 2y + y' = 4x + 2
Initial condition: y(-1) = e² - 2
Solution candidate: y = e^(-2x) + 2x
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.38
Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.
Verified step by step guidance1
Start with the given function: \(f(x) = mx + b\), where \(m\) and \(b\) are constants and \(m \neq 0\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = mx + b\).
Swap the roles of \(x\) and \(y\) to find the inverse: \(x = my + b\).
Solve this equation for \(y\) to express the inverse function: subtract \(b\) from both sides to get \(x - b = my\), then divide both sides by \(m\) to isolate \(y\): \(y = \frac{x - b}{m}\).
Rewrite the inverse function in slope-intercept form: \(f^{-1}(x) = \frac{1}{m}x - \frac{b}{m}\). This shows the inverse is a line with slope \(\frac{1}{m}\) and \(y\)-intercept \(-\frac{b}{m}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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 a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x. Understanding how to find and interpret inverses is essential for analyzing the relationship between a function and its inverse graphically.
Recommended video:
4:49
Inverse Cosine
Linear Functions and Their Properties
A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept. The graph is a straight line, and the slope indicates the rate of change. Recognizing how slope and intercept affect the line helps in determining the characteristics of its inverse.
Recommended video:
06:21Properties of Functions
Finding the Inverse of a Linear Function
To find the inverse of f(x) = mx + b, solve for x in terms of y, then interchange variables. This process yields f⁻¹(x) = (1/m)x - b/m, showing the inverse is also linear with slope 1/m and y-intercept -b/m. This algebraic manipulation is key to proving the inverse's graph properties.
Recommended video:
07:17Linearization
Related Practice
Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)
1
views
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(5-7x)
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
29. y = (1 - t)coth⁻¹(√t)
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))