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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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).

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.

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.

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