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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 7.1.43a
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 guidance
1
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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Linearization

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

In Exercises 41–44:

a. Find f⁻¹(x).


42. f(x) = (x + 2) / (1 − x), a = 1/2

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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)

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Textbook Question

89. Use limits to find horizontal asymptotes for each function.

a. y = x tan(1/x)

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Textbook Question

145. The linearization of eˣ at x = 0

a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.

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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?

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