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

Guida verificata passo dopo passo

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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