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Section 3.6 : Combining Functions

2. Given \(P\left( t \right) = 4{t^2} + 3t - 1\) and \(A\left( t \right) = 2 - {t^2}\) compute each of the following.

  1. \(\left( {P + A} \right)\left( t \right)\)
  2. \(A - P\)
  3. \(\left( {PA} \right)\left( t \right)\)
  4. \(\displaystyle \left( {\frac{P}{A}} \right)\left( { - 2} \right)\)

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a \(\left( {P + A} \right)\left( t \right)\) Show Solution

Not much to do here other than do the evaluation. Also, remember that in this case the “evaluation” really only consists of plugging the two equations for each function in and doing some basic algebraic manipulations to simplify the answer if possible.

\[\left( {P + A} \right)\left( t \right) = P\left( t \right) + A\left( t \right) = 4{t^2} + 3t - 1 + 2 - {t^2} = \require{bbox} \bbox[2pt,border:1px solid black]{{3{t^2} + 3t + 1}}\]

b \(A - P\) Show Solution

Not much to do here other than do the evaluation. Also, remember that in this case the “evaluation” really only consists of plugging the two equations for each function in and doing some basic algebraic manipulations to simplify the answer if possible.

\[A - P = A\left( t \right) - P\left( t \right) = 2 - {t^2} - \left( {4{t^2} + 3t - 1} \right) = 2 - {t^2} - 4{t^2} - 3t + 1 = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 5{t^2} - 3t + 3}}\]

c \(\left( {PA} \right)\left( t \right)\) Show Solution

Not much to do here other than do the evaluation. Also, remember that in this case the “evaluation” really only consists of plugging the two equations for each function in and doing some basic algebraic manipulations to simplify the answer if possible.

\[\begin{align*}\left( {PA} \right)\left( t \right) = P\left( t \right)A\left( t \right) &= \left( {4{t^2} + 3t - 1} \right)\left( {2 - {t^2}} \right)\\ & = 8{t^2} - 4{t^4} + 6t - 3{t^3} - 2 + {t^2} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 4{t^4} - 3{t^3} + 9{t^2} + 6t - 2}}\end{align*}\]

For the product just remember to distribute every term from the first polynomial through the second polynomial and the combine like terms to simplify.


d \(\displaystyle \left( {\frac{P}{A}} \right)\left( { - 2} \right)\) Show Solution

Not much to do here other than do the evaluation. We’ll leave it to you to verify the specific function evaluations.

\[\left( {\frac{P}{A}} \right)\left( { - 2} \right) = \frac{{P\left( { - 2} \right)}}{{A\left( { - 2} \right)}} = \frac{9}{{ - 2}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{9}{2}}}\]