Question

the function ( s(t) = 4t - 21 ) is a result of the composition ( (q circ p)(t) ). if ( q(t) = 4t^3 - 1 ), what is ( p(t) )?
( p(t) = t - 20 )
( p(t) = sqrt3{t - 5} )
( p(t) = sqrt3{t} - 20 )
( p(t) = sqrt{t - 5} )

Explanation:

Step1: Define composition

$(q \circ p)(t) = q(p(t)) = 4t - 21$

Step2: Substitute $q(t)$ formula

$4[p(t)]^3 - 1 = 4t - 21$

Step3: Isolate the cubic term

$4[p(t)]^3 = 4t - 21 + 1 = 4t - 20$

Step4: Solve for $[p(t)]^3$

$[p(t)]^3 = \frac{4t - 20}{4} = t - 5$

Step5: Take cube root

$p(t) = \sqrt[3]{t - 5}$

Answer:

$p(t) = \sqrt[3]{t - 5}$