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Chapter 1: Problem 34

Find (a) \(f \circ g,\) (b) \(g \circ f,\) and (c) \(g \circ g\). $$f(x)=x^{3}, \quad g(x)=\frac{1}{x}$$

Short Answer

Expert verified
The function compositions are \(f \circ g(x) = \frac{1}{x^{3}}\), \(g \circ f(x) = \frac{1}{x^{3}}\) and \(g \circ g(x) = x\).

Step by step solution

01

Find \(f \circ g\)

To find \(f \circ g(x)\), substitute \(g(x)\) in place of \(x\) in function \(f(x)\). So, \(f \circ g(x) = f(g(x))\), Substituting \(g(x)\) into \(f\), we get, \(f(\frac{1}{x})\). Replacing \(x\) with \(\frac{1}{x}\) in \(f(x)=x^{3}\), you get \(f(\frac{1}{x}) = (\frac{1}{x})^{3} = \frac{1}{x^{3}}\).
02

Find \(g \circ f\)

Similarly to find \(g \circ f(x)\), substitute \(f(x)\) in place of \(x\) in function \(g(x)\). So, \(g \circ f(x) = g(f(x))\), Substituting \(f(x)\) into \(g\), we get, \(g(x^{3})\). Replacing \(x\) with \(x^{3}\) in \(g(x)=\frac{1}{x}\), you get \(g(x^{3}) = \frac{1}{x^{3}} = \frac{1}{x^{3}}\).
03

Find \(g \circ g\)

To find \(g \circ g(x)\), substitute \(g(x)\) in place of \(x\) in function \(g(x)\) again. So, \(g \circ g(x) = g(g(x))\), substituting \(g(x)\) into \(g\), we get, \(g(\frac{1}{x})\). Replacing \(x\) with \(\frac{1}{x}\) in \(g(x)=\frac{1}{x}\), you get \(g(\frac{1}{x}) = \frac{1}{(\frac{1}{x})} = x\).

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