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Chapter 1: Problem 43

Let \(f(x)=|x|, g(x)=x^{2}-4\) \(F(x)=\sqrt{x},\) and \(G(x)=1 /(x-2) .\) Determine the following composite functions and give their domains. $$f \circ G$$

Short Answer

Expert verified
Question: Determine the composite function \(f \circ G(x)\) and its domain given the functions \(f(x) = |x|\) and \(G(x) = \frac{1}{x-2}\). Answer: The composite function \(f \circ G(x) = \left|\frac{1}{x-2}\right|\) and its domain is \((-\infty, 2) \cup (2, \infty)\).

Step by step solution

01

Identify the functions

In this problem, we are given: - \(f(x) = |x|\) - \(G(x) = \frac{1}{x-2}\) We are asked to find the composite function \(f \circ G(x)\).
02

Determine the composite function

To find the composite function \(f \circ G(x)\), we need to plug in the function \(G(x)\) into the function \(f(x)\): \(f \circ G(x) = f(G(x)) = |G(x)|\) Now, we plug in the expression of G(x) into the expression for f(x): \(f \circ G(x) = \left|\frac{1}{x-2}\right|\)
03

Determine the domain of the composite function

To find the domain of the composite function, we need to analyze the function. Recall that the domain consists of all possible input values of x for which the function is defined. We know that the absolute value function is defined for all real numbers. However, we have a fraction within the absolute value bars, and the denominator cannot be equal to zero. Therefore, we must exclude any value of x that would make the denominator of the fraction equal to zero: \(x-2 \neq 0\) Solving for x, we find: \(x \neq 2\) Thus, the domain of the composite function \(f \circ G(x)\) is all real numbers except for 2, which can be expressed as: Domain = \((-\infty, 2) \cup (2, \infty)\)

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