Textbook Question
Finding all inverses Find all the inverses associated with the following functions, and state their domains.
ƒ(x) = 2 / ( x² + 2)
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 1.54
More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.
G o G
Verified step by step guidance1
Step 1: Understand the notation \( G \circ G \). This represents the composition of the function \( G(x) = \frac{1}{x-2} \) with itself, meaning \( G(G(x)) \).
Step 2: Substitute \( G(x) \) into itself. Start by replacing \( x \) in \( G(x) \) with \( G(x) \). This gives \( G(G(x)) = G\left(\frac{1}{x-2}\right) = \frac{1}{\left(\frac{1}{x-2}\right) - 2} \).
Step 3: Simplify the expression \( \frac{1}{\left(\frac{1}{x-2}\right) - 2} \). To do this, find a common denominator for the terms in the denominator: \( \frac{1}{x-2} - 2 = \frac{1 - 2(x-2)}{x-2} = \frac{1 - 2x + 4}{x-2} = \frac{5 - 2x}{x-2} \).
Step 4: The expression becomes \( G(G(x)) = \frac{x-2}{5-2x} \).
Step 5: Determine the domain of \( G(G(x)) \). The domain of \( G(G(x)) \) is all real numbers except where the denominator is zero. Solve \( 5 - 2x = 0 \) to find the values to exclude from the domain. Also, consider the domain of \( G(x) \) itself, which excludes \( x = 2 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composite Functions
A composite function is formed when one function is applied to the result of another function. It is denoted as (f o g)(x) = f(g(x)). Understanding how to combine functions is essential for evaluating composite functions, as it requires substituting the output of one function into another.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with composite functions, it is crucial to determine the domain of each individual function and how they interact, as the domain of the composite function may be restricted by the domains of the functions involved.
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Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is important in the context of composite functions, as it can affect the overall behavior and domain of the resulting composite function, particularly when combined with other functions that may have restrictions.
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Related Practice
Textbook Question
Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.
logb x/y
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Textbook Question
Sketch the graph of the inverse of ƒ. <IMAGE>
Textbook Question
Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .
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Textbook Question
Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.
cos⁻¹ (- 1/2 )
Textbook Question
The National Weather Service releases approximately radiosondes every year to collect data from the atmosphere. Attached to a balloon, a radiosonde rises at about ft/min until the balloon bursts in the upper atmosphere. Suppose a radiosonde is released from a point ft above the ground and that seconds later, it is ft above the ground. Let represent the height (in feet) that the radiosonde is above the ground seconds after it is released. Evaluate and interpret the meaning of this quotient.