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Chapter 1: Problem 38

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=x^{2 / 3}, \quad g(x)=x^{6}$$

Short Answer

Expert verified
The function \(f \circ g(x)\) is \(x^{4}\) and its domain is all real numbers. The function \(g \circ f(x)\) is also \(x^{4}\) and its domain is all real numbers.

Step by step solution

01

Compute f(g(x))

Substituting \(g(x)\) into \(f(x)\), we get \(f(g(x)) = [(x^6)^{2/3}]\). Simplifying this equation, we end up with \(x^{4}\).
02

Compute g(f(x))

Substituting \(f(x)\) into \(g(x)\), we get \(g(f(x)) = [(x^{2/3})^6]\). Simplifying this equation, we get \(x^{4}\).
03

Find the domain of each function and each composite function

The domain of \(f(x)\) is all real numbers because we can take the cube root of any number and then square it. Similarly, the domain of \(g(x)\) is all real numbers because we can raise any real number to the sixth power. The domains of the composite functions \(f(g(x))\) and \(g(f(x))\) are also all real numbers because both composite functions are equal to \(x^{4}\), which is defined for all real numbers.

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