Question

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

\(R\left( x \right) = \sqrt {\sqrt x - 1} \)

Short answer

The function \(R\left( x \right) = \sqrt {\sqrt x - 1} \)can be expressed in the form \(f \circ g\)with\(f\left( x \right) = \sqrt x \),\(g\left( x \right) = x - 1\) and \(h\left( x \right) = \sqrt x \).

Step-by-step solution

Given data

The provided function is

\(R\left( x \right) = \sqrt {\sqrt x - 1} \)

For three functions \(f\left( x \right)\), \(g\left( x \right)\)and\(h\left( x \right)\)their composition is defined as

\(f \circ g \circ h = f\left( {g\left( {h\left( x \right)} \right)} \right)\;\;\;\;\;.....\left( 1 \right)\)

Write the function

Let \(f\left( x \right) = \sqrt x \), \(g\left( x \right) = x - 1\) and \(h\left( x \right) = \sqrt x \)

From equation (1),

\(\begin{aligned}f \circ g \circ h &= f\left( {g\left( {\sqrt x } \right)} \right)\\ &= f\left( {\sqrt x - 1} \right)\\ &= \sqrt {\sqrt x - 1} \\ &= R\left( x \right)\end{aligned}\)

Hence, the required functions are ,\(g\left( x \right) = x - 1\) and \(h\left( x \right) = \sqrt x \).