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Chapter 0: Problem 1

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ f(x)=\frac{1}{3} x^{3} ; \quad g(x)=\sqrt[3]{3 x} $$

Short Answer

Expert verified
We showed that \(f[g(x)] = x\) and \(g[f(x)] = x\) by plugging in the functions \(f(x) = \frac{1}{3}x^3\) and \(g(x) = \sqrt[3]{3x}\) into each other and simplifying the resulting expressions. Since both conditions are met, functions f and g are inverses of each other.

Step by step solution

01

Find f[g(x)]

First, we need to find f[g(x)] by plugging g(x) into the function f(x). Given the functions f(x) = \(\frac{1}{3}x^3\) and g(x) = \( \sqrt[3]{3x}\), we can write: \(f[g(x)] = \frac{1}{3}(\sqrt[3]{3x})^3 \)
02

Simplify f[g(x)]

Now, let's simplify the expression to see if it equals x: \(f[g(x)] = \frac{1}{3}(\sqrt[3]{3x})^3 = \frac{1}{3}(3x) = x \) As we can see, f[g(x)] simplifies to x.
03

Find g[f(x)]

Next, we need to find g[f(x)] by plugging f(x) into the function g(x). Using the functions f(x) = \(\frac{1}{3}x^3\) and g(x) = \(\sqrt[3]{3x}\), we can write: \(g[f(x)] = \sqrt[3]{3(\frac{1}{3} x^3)} \)
04

Simplify g[f(x)]

Finally, let's simplify the expression to see if it equals x: \(g[f(x)] = \sqrt[3]{3(\frac{1}{3} x^3)} = \sqrt[3]{x^3} = x \) As we can see, g[f(x)] also simplifies to x. Since both \(f[g(x)] = x\) and \(g[f(x)] = x\), we can conclude that functions f and g are indeed inverses of each other.

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