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Chapter 1: Problem 35

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=\sqrt{x+4}, \quad g(x)=x^{2}$$

Short Answer

Expert verified
The composite functions are: \(f \circ g = \sqrt{x^{2}+4}\) with a domain of (-鈭�, 鈭�), and \(g \circ f = x+4\) with a domain \(x \geq -4\).

Step by step solution

01

Compute Composite Functions

To compute the composite function \(f(g(x))\) which is the same as \(f \circ g\), substitute \(x^{2}\) (which is g(x)) into f. Thus, \(f(g(x)) = f(x^{2}) = \sqrt{x^{2}+4}\). However, for \(g(f(x))\) which is the same as \(g \circ f\), substitute \(\sqrt{x+4}\) (which is f(x)) into g, and get \(g(f(x)) = g(\sqrt{x+4}) = (\sqrt{x+4})^{2}\).
02

Simplify the Composite Functions

For \(f \circ g\) we can't further simplify, so \(f \circ g = \sqrt{x^{2}+4}\). For \(g \circ f\), \(g(\sqrt{x+4}) = (\sqrt{x+4})^{2}\), we can simplify it further and get \(x+4.\)
03

Find the Domain of the Functions and Composite Functions

The domain of the function \(f(x) = \sqrt{x+4}\) is \(x \geq -4\), since, for real numbers, we can鈥檛 take the square root of a negative number. The domain of \(g(x) = x^{2}\) is (-鈭�, 鈭�), since all real numbers can be squared. For the composite function \(f \circ g = \sqrt{x^{2}+4}\), the domain is (-鈭�, 鈭�). For the composite function \(g \circ f = x+4\), the domain is \(x \geq -4\), since the function inside the square, which is \(\sqrt{x+4}\), needs \(x \geq -4\) to be real.

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