Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.1.39b

Chapter 7, Problem 7.1.39b

Find the inverse of f(x)=x+b (b constant). How is the graph of f^(-1) related to the graph of f?

Verified step by step guidance

Start with the function definition: \(f(x) = x + b\), where \(b\) is a constant.

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x + b\).

Swap the variables \(x\) and \(y\) to find the inverse: \(x = y + b\).

Solve this equation for \(y\) to express the inverse function: \(y = x - b\).

Understand the graphical relationship: the graph of \(f^{-1}\) is the reflection of the graph of \(f\) across the line \(y = x\).

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Key Concepts

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

Inverse Function

An inverse function reverses the effect of the original function, mapping outputs back to their inputs. For f(x) = x + b, the inverse function f⁻¹(x) solves for x in terms of y, effectively undoing the addition of b.

Finding the Inverse Algebraically

To find the inverse, replace f(x) with y, swap x and y, then solve for y. For f(x) = x + b, swapping gives x = y + b, so solving for y yields f⁻¹(x) = x - b.

Graphical Relationship Between a Function and Its Inverse

The graph of an inverse function is the reflection of the original function's graph across the line y = x. This symmetry means points (a, b) on f correspond to points (b, a) on f⁻¹.

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b. How long will it take to eradicate the disease—that is, reduce the number of cases to less than 1?

Textbook Question

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

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Textbook Question

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b. ln 9.8

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Textbook Question

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Textbook Question

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

2. y' = y²

b. y = -1/(x+3)