Textbook Question
Find the inverse of the function f(x)=mx, where m is a constant different from zero.
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 7.1.43a
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
Verified step by step guidance1
Start by writing the function given: \(f(x) = 5 - 4x\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = 5 - 4x\).
Swap the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = 5 - 4y\).
Solve this equation for \(y\) to express \(y\) in terms of \(x\): subtract 5 from both sides to get \(x - 5 = -4y\), then divide both sides by \(-4\) to isolate \(y\).
Write the inverse function as \(f^{-1}(x) = \frac{5 - x}{4}\) (after simplifying the expression).
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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. If f(x) maps x to y, then f⁻¹(x) maps y back to x. Finding the inverse involves solving the equation y = f(x) for x in terms of y.
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Solving Linear Equations
Since f(x) = 5 − 4x is linear, finding its inverse requires isolating x in the equation y = 5 − 4x. This involves algebraic manipulation such as adding, subtracting, multiplying, or dividing both sides to solve for x.
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Domain and Range Considerations
When finding an inverse, it is important to consider the domain and range of the original function and its inverse. The value a = 1/2 may specify a particular input or output to check, ensuring the inverse function is valid at that point.
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Related Practice
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
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Textbook Question
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?
Textbook Question
a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
Textbook Question
77. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 3) (x - 3) / (x² - 3) = lim (x → 3) 1 / (2x) = 1/6