Question

c(t) = (t^2, pi*t)
g(u,v) = (e^u, u+sin(v))
f(x,y) = (xy, x^2, 3y)

What describes the image in R^3 (f o g o c) for all t in R? A point, a curve, a surface or a solid?

Answer 1

I think its a curve.

Answer 2

I'm inclined to agree with you, but don't take my word for it.

\[(f\circ g\circ c)(t)=\bigg(e^{t^2}\left(t^2+\sin\pi t\right),~e^{2t^2},~3t^2+3\sin\pi t\bigg)\]
It's obviously not a point, since \((f\circ g\circ c)(a)\not=(f\circ g\circ c)(b)\) for all \(a,b\in\mathbb{R}\), so you can eliminate that option.

I'm a bit iffy on the reasoning for why it's not a solid/surface, but only because I've dealt with solids/surfaces using functions like \(f:\mathbb{R}^3\to\mathbb{R}\) and \(f:\mathbb{R}^2\to\mathbb{R}\), respectively.