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Numerade Educator

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Problem 47 Medium Difficulty

Express the function in the form $ f \circ g $.

$ v(t) = \sec (t^2) \tan (t^2) $

Answer

$$\text { Let } g(t)=t^{2} \text { and } f(t)=\sec t \tan t . \text { Then }(f \circ g)(t)=f(g(t))=f\left(t^{2}\right)=\sec \left(t^{2}\right) \tan \left(t^{2}\right)=v(t)$$.

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Video Transcript

here we have a function V of tea and we want to think of V A. T as a composition of functions f of G and another way to write that would be f of G of tea with parentheses. And looking at that notation, we can see that g of t is the inside function and f of tea is the outside function. So thinking about it that way we could see that the inside function could be t squared so we could let g of t equal t squared. And so the outside function would be C can't t tangent t And that way, if we found f of G, we could substitute t squared in for tea and we would get the original function v of tea.