Question

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

46. \(F\left( x \right) = {\bf{co}}{{\bf{s}}^{\bf{2}}}x\)

Short answer

The function is \(f \circ g\left( x \right) = {\cos ^2}x = F\left( x \right)\).

Step-by-step solution

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

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

\(g\left( x \right) = \cos x\) and \(f\left( x \right) = {x^2}\).

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

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

\(\begin{aligned}f \circ g\left( x \right) &= f\left( {g\left( x \right)} \right)\\ &= f\left( {\cos x} \right)\\ &= {\left( {\cos x} \right)^2}\\ &= {\cos ^2}x\end{aligned}\)

So, the required functions are \(f\left( x \right) = {x^2}\) and \(g\left( x \right) = \cos x\).