Question

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

\(u\left( t \right) = \frac{{\tan t}}{{1 + \tan t}}\)

Short answer

The function \(u\left( t \right) = \frac{{\tan t}}{{1 + \tan t}}\)can be expressed in the form \(f \circ g\)with \(f\left( t \right) = \frac{t}{{1 + t}}\) and \(g\left( t \right) = \tan t\).

Step-by-step solution

Given data

The provided function is \(u\left( t \right) = \frac{{\tan t}}{{1 + \tan t}}\).

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

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

Write the Function

Let \(f\left( t \right) = \frac{t}{{1 + t}}\) and \(g\left( t \right) = \tan t\)

From equation (1),

\(\begin{aligned}f \circ g &= f\left( {\tan t} \right)\\ &= \frac{{\tan t}}{{1 + \tan t}}\\ &= u\left( t \right)\end{aligned}\)

Hence, the required functions are \(f\left( t \right) = \frac{t}{{1 + t}}\) and \(g\left( t \right) = \tan t\).