Textbook Question
Solve the following equations.
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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 47
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.
ƒ o g
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Identify the composite function \( (f \circ g)(x) \), which means \( f(g(x)) \).
Substitute \( g(x) = x^2 - 4 \) into \( f(x) = |x| \), so \( f(g(x)) = |x^2 - 4| \).
Determine the domain of \( g(x) = x^2 - 4 \), which is all real numbers since a polynomial is defined everywhere.
Since \( f(x) = |x| \) is also defined for all real numbers, the domain of \( f(g(x)) = |x^2 - 4| \) is all real numbers.
Conclude that the composite function \( (f \circ g)(x) = |x^2 - 4| \) is defined for all real numbers.
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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 g(x) into f(x).
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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 both the inner and outer functions, as the overall domain is influenced by any restrictions from either function.
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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 because it can affect the overall behavior and domain of the composite function, particularly when combined with other functions that may have their own restrictions.
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Related Practice
Textbook Question
Solve the following equations.
Textbook Question
Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.
a. Find A(2) .
ƒ(t) =6 <IMAGE>
Textbook Question
Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.
b. Find A(6).
ƒ(t) =6 <IMAGE>
Textbook Question
Slope functions Determine the slope function S (x) for the following functions
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Textbook Question
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.
ƒ o G