Composite functions

[tymom]

I need help getting started with the following word problem. Please advise on how to setup the equation
The volume V of a spherical balloon with radius r is given by V(r)=4/3π r³suppose that the balloon is being inflated so that the radius in inches after t seconds is r(t) =∛(t).

(a) Evaluate (V*r)(3) and interpret your result.
b) Find (V*r)(t).

 

[JeffM]

I need help getting started with the following word problem. Please advise on how to setup the equation
The volume V of a spherical balloon with radius r is given by V(r)=4/3π r³suppose that the balloon is being inflated so that the radius in inches after t seconds is r(t) =∛(t).

(a) Evaluate (V*r)(3) and interpret your result. I don't think this is what you mean. Please check the problem carefully.
b) Find (V*r)(t). I don't think this is what you mean. Please check the problem carefully.

V(r) is a rule that takes r as an input and produces V(r) as an output.

r(t) is a rule that takes t as an input and produces r(t) as an output.

V(r(t)) is a rule that says take t as the input to rule r(t) and then take the output of that rule as the input to rule V(r). A composite function just tells you to put two or more rules together in a chain, starting from the inside out.

\(\displaystyle So\ V(r) = \frac{4}{3} \pi r^3\ and\ r(t) = r = \sqrt[3]{t} \implies V(r(t)) = V(\sqrt[3]{t}) = \frac{4}{3} \pi \left(\sqrt[3]{t}\right)^3 =\)

Can you finish it up?

 

[mmm4444bot]

HERE is a page with a graphic depiction of function composition, using the "function as machine" model. :cool:

 

[mmm4444bot]

ֺ

Oh, I forgot to mention: type a lower-case o instead of * to show composition.

Symbol * is already used to show multiplication.

"f of g (of x)" is texted as fog(x) … also some prefer (fog)(x) or even f{g(x)}

Hence, "V of r (of t)" is texted as Vor(t) or (Vor)(t), et cetera

Cheers :cool:

 

[Deleted member 4993]

ֺ

Oh, I forgot to mention: type a lower-case o instead of * to show composition.

Symbol * is already used to show multiplication.

"f of g (of x)" is texted as fog(x) … also some prefer (fog)(x) .... I prefer f{g(x)}

"V of r (of t)" is texted as Vor(t) or (Vor)(t)

Cheers :cool:

.