Question

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

\(v\left( t \right) = \sec \left( {{t^2}} \right)\tan \left( {{t^2}} \right)\)

Short answer

The function\(v\left( t \right) = \sec \left( {{t^2}} \right)\tan \left( {{t^2}} \right)\)can be expressed in the form \(f \circ g\)with \(f\left( t \right) = \sec \left( t \right)\tan \left( t \right)\) and\(g\left( t \right) = {t^2}\).

Step-by-step solution

Given data

The provided function is

\(v\left( t \right) = \sec \left( {{t^2}} \right)\tan \left( {{t^2}} \right)\)

For two functions \(f\left( t \right)\) and\(g\left( t \right)\)their composition is defined as

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

Step 2:Write the function

Let \(f\left( t \right) = \sec t\tan t\) and \(g\left( t \right) = {t^2}\)

From equation (1),

\(\begin{aligned}f \circ g &= f\left( {{t^2}} \right)\\ &= \sec \left( {{t^2}} \right)\tan \left( {{t^2}} \right)\\ &= v\left( t \right)\end{aligned}\)

Hence, the required functions are \(f\left( t \right) = \sec t\tan t\)and\(g\left( t \right) = {t^2}\).