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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 47
Capitolo 1, Problema 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

Guida verificata passo dopo passo
1
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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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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Evaluate Composite Functions - Special Cases

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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Finding the Domain and Range of a Graph

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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Initial Value Problems
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Domanda del libro di testo

Solve the following equations.


{Use of Tech} sin2θ=15,0<θ<π2\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\)\(\lt{\frac{\pi}{2}\)}

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Solve the following equations.

tan22θ=1,0<θ<π\(\tan\)^22\(\theta\)=1,0\(\lt{\theta}\)\(\lt{\pi}\)

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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>

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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>

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Slope functions Determine the slope function S (x) for the following functions

ƒ(x)=∣x∣ƒ(x) = | x |

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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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