Question

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

50. \(H\left( x \right) = \sqrt {{\bf{1}} + \sqrt x } \)

Short answer

The function is \(f \circ g\left( t \right) = \sqrt {1 + \sqrt x } \).

Step-by-step solution

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

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

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

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

The composite function \(f \circ g\) can be obtained as:

\(\begin{aligned}f \circ g\left( t \right) &= f\left( {g\left( x \right)} \right)\\ &= f\left( {\sqrt x } \right)\\ &= \sqrt {1 + \sqrt x } \end{aligned}\)

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