Section 3.6 : Combining Functions
5. Given \(R\left( t \right) = \sqrt t - 2\) and \(A\left( t \right) = {\left( {t + 2} \right)^2}\), \(t \ge 0\) compute each of the following.
- \(\left( {R \circ A} \right)\left( t \right)\)
- \(\left( {A \circ R} \right)\left( t \right)\)
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a \(\left( {R \circ A} \right)\left( t \right)\) Show SolutionRemember that the “\( \circ \)” denotes composition and not multiplication!
\[\left( {R \circ A} \right)\left( t \right) = R\left[ {A\left( t \right)} \right] = R\left[ {{{\left( {t + 2} \right)}^2}} \right] = \sqrt {{{\left( {t + 2} \right)}^2}} - 2 = \left( {t + 2} \right) - 2 = \require{bbox} \bbox[2pt,border:1px solid black]{t}\]b \(\left( {A \circ R} \right)\left( t \right)\) Show Solution
Remember that the “\( \circ \)” denotes composition and not multiplication!
\[\left( {A \circ R} \right)\left( t \right) = A\left[ {R\left( t \right)} \right] = A\left[ {\sqrt t - 2} \right] = {\left( {\sqrt t - 2 + 2} \right)^2} = {\left( {\sqrt t } \right)^2} = \require{bbox} \bbox[2pt,border:1px solid black]{t}\]