Question

Express the function in the form \(f \circ g\).

47. \(F\left( x \right) = \frac{{\sqrt[{\bf{3}}]{x}}}{{{\bf{1}} + \sqrt[{\bf{3}}]{x}}}\)

Short answer

The function is \(f \circ g\left( x \right) = \frac{{\sqrt[3]{x}}}{{1 + \sqrt[3]{x}}} = F\left( x \right)\).

Step-by-step solution

Find the constituent functions for \(F\left( x \right)\)

If \(F\left( x \right) = f\left[ {g\left( x \right)} \right]\), then functions \(f\left( x \right)\) and \(g\left( x \right)\) can be expressed as,

\(g\left( x \right) = \sqrt[3]{x}\) and \(f\left( x \right) = \frac{x}{{1 + x}}\).

Verify the given composite function using \(f\left( x \right)\) and \(g\left( x \right)\)

Thecomposite function\(f \circ g\) can be obtained as shown below:

\(\begin{aligned}f \circ g\left( x \right) &= f\left[ {g\left( x \right)} \right]\\ &= f\left( {\sqrt[3]{x}} \right)\\ &= \frac{{\sqrt[3]{x}}}{{1 + \sqrt[3]{x}}}\end{aligned}\)

So, the required functions are \(f\left( x \right) = \frac{x}{{1 + x}}\) and \(g\left( x \right) = \sqrt[3]{x}\).