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 o ƒ

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Step 1: Understand the notation G $\circ$ g $\circ$ ƒ. This represents the composition of functions, which means you apply the functions from right to left. Start with ƒ(x), then apply g to the result, and finally apply G.

Step 2: Begin with ƒ(x) = |x|. This function takes any real number x and returns its absolute value.

Step 3: Apply g(x) = x^2 - 4 to the result of ƒ(x). This means you substitute |x| into g, resulting in g(ƒ(x)) = (|x|)^2 - 4.

Step 4: Now apply G(x) = $\frac{1}{x-2}$ to the result of g(ƒ(x)). Substitute (|x|)^2 - 4 into G, giving G(g(ƒ(x))) = $\frac{1}{(|x|)^2 - 4 - 2}$.

Step 5: Determine the domain of the composite function G(g(ƒ(x))). The expression (|x|)^2 - 4 must not equal 2 to avoid division by zero. Solve (|x|)^2 - 4 $\neq$ 2 to find the domain.

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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. For example, if we have functions f and g, the composite function g(f(x)) means we first apply f to x and then apply g to the result. Understanding how to combine functions is essential for solving problems involving multiple functions.

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 output of one function becomes the input for the next. This ensures that all operations within the composite function are valid.

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 composite functions because it can affect the overall behavior and domain of the resulting function. For instance, when combined with other functions, the absolute value can introduce restrictions on the input values that must be considered.

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