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Numerade Educator

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Problem 45 Medium Difficulty

Express the function in the form $ f \circ g $.

$ F(x) = \dfrac{\sqrt[3]{x}}{1 + \sqrt[3]{x}} $

Answer

$$\text { Let } g(x)=\sqrt[3]{x} \text { and } f(x)=\frac{x}{1+x} . \text { Then }(f \circ g)(x)=f(g(x))=f(\sqrt[3]{x})=\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}=F(x)$$.

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Video Transcript

here we have capital F of X and we were told to think of capital F of X as the composition f of g. Another way to write that is with parentheses f of G of X. And if we do that, we can tell that G of X is the inside function and f of X is the outside function. So looking back at what we were given, we could say that the cube root of X is our inside function. It happens to appear twice so we could say that G of X is the cube root of X. Now, what was that put inside of that was put inside of F. So what would I have to look like? It would have X on the top and one plus X on the bottom. That way you could substitute the G function into both places.