Question

use the variable u for the inside function to express each of the following as a composite function.
(a) ( y = left( 3 t^2 - 1
ight)^2 )
note: ( u(t) ) is the inside function and ( y(u) ) is the outside function.
( u(t) = )
( y(u) = )
(b) ( p = 7 e^{-0.2t} )
note: ( u(t) ) is the inside function and ( p(u) ) is the outside function.
( u(t) = )
( p(u) = )
(c) ( c = 2 lnleft( q^5 + 9
ight) )
note: ( u(q) ) is the inside function and ( c(u) ) is the outside function.
( u(q) = )
( c(u) = )

Explanation:

Part (a)

Step1: Identify inside function

The inside function \( u(t) \) is the part inside the parentheses. So \( u(t) = 3t^2 - 1 \).

Step2: Identify outside function

The outside function \( y(u) \) takes \( u \) and squares it. So \( y(u) = u^2 \).

Step1: Identify inside function

The inside function \( u(t) \) is the exponent of \( e \). So \( u(t) = -0.2t \).

Step2: Identify outside function

The outside function \( P(u) \) takes \( u \), computes \( e^u \), and multiplies by 7. So \( P(u) = 7e^u \).

Step1: Identify inside function

The inside function \( u(q) \) is the argument of the natural logarithm. So \( u(q) = q^5 + 9 \).

Step2: Identify outside function

The outside function \( C(u) \) takes \( u \), computes \( \ln(u) \), and multiplies by 2. So \( C(u) = 2\ln(u) \).

Answer:

\( u(t) = 3t^2 - 1 \)
\( y(u) = u^2 \)

Part (b)