Question

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

53. \(S\left( t \right) = {\sin ^{\bf{2}}}\left( {\cos t} \right)\)

Short answer

The function is expressed as \(\left( {f \circ g \circ h} \right)\left( t \right) = {\sin ^2}\left( {\cos t} \right) = S\left( t \right)\).

Step-by-step solution

The composite function

Consider two functions \(f\) and \(g\), the composite function\(f \circ g\)(also called the composition of \(f\) and \(g\)) is defined as:

\(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\)

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

Consider the functions as \(h\left( t \right) = \cos t,{\rm{ }}g\left( t \right) = \sin t,\,\,{\mathop{\rm and}\nolimits} \,\,f\left( t \right) = {t^2}\).

Express the function in the form \(f \circ g \circ h\) as shown below:

\(\begin{aligned}\left( {f \circ g \circ h} \right)\left( t \right) &= f\left( {g\left( {h\left( t \right)} \right)} \right)\\ &= f\left( {g\left( {\cos t} \right)} \right)\\ &= f\left( {\sin \left( {\cos t} \right)} \right)\\ &= {\left( {\sin \left( {\cos t} \right)} \right)^2}\\ &= {\sin ^2}\left( {\cos t} \right)\\ &= S\left( t \right)\end{aligned}\)

Thus, it is obtained that \(\left( {f \circ g \circ h} \right)\left( t \right) = {\sin ^2}\left( {\cos t} \right) = S\left( t \right)\).