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Chapter 1: Problem 42

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. $$g \circ f$$

Short Answer

Expert verified
Question: Find the composite function \(g \circ f\) and its domain given that \(f(x) = |x|\) and \(g(x) = x^2 - 4\). Answer: The composite function \(g \circ f\) is \(g(f(x)) = x^2 - 4\), and its domain is all real numbers.

Step by step solution

01

Determine the composite function \(g \circ f\)

To find the composite function \(g \circ f\), we need to substitute the function \(f(x)\) into the function \(g(x)\). So we have \(g(f(x)) = g(|x|) = (|x|)^2 - 4\).
02

Simplify the expression

Now simplify the function \(g(f(x))\). Since the square of the absolute value is equal to the square of the original value, we can rewrite the function as, \(g(f(x)) = (|x|)^2 - 4 = x^2 - 4\).
03

Determine the domain of the composite function

Since the composite function is just a quadratic equation, its domain is all real numbers because we can plug in any real number for \(x\) and the function will be defined. Therefore, the domain of \(g \circ f\) is \(\mathbb{R}\). In conclusion, the composite function \(g \circ f\) is \(g(f(x)) = x^2 -4\), and its domain is all real numbers.

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