When defining the tangent space there is some function $f$ that is defined on the manifold in order for the vector (as it is a linear map) to act on. I'm confused as to what this function does and what an example would be.
Say we have some manifold $M$ and a family of curves, parameterised by $\lambda$, going through a point $p$ on the manifold.
$\phi : \mathbb{R}\rightarrow M$
$\phi :\lambda \mapsto \phi(\lambda) $.
$\phi(0) = p$.
We have some chart map $x$ that takes a local region into $\mathbb{R}^n$ and we name the image of the curves in the chart $x^\mu(\lambda)$. $x : M \rightarrow \mathbb{R}^n$
$(x\circ\phi) : \mathbb{R}\rightarrow M \rightarrow \mathbb{R}^n$
$(x\circ\phi): \lambda\mapsto \phi(\lambda) \mapsto x^\mu(\lambda)$
Then we define some functions on the manifold. $f : M \rightarrow \mathbb{R}$, where $f \in C^\infty(M)$. Then we evaluate the function $f$ on the curve on the manifold, $f(\phi(\lambda))$.
The tangent vector acting on $f$ to the curve at the point $p$ is: $$V_\phi(f) = \frac{\partial}{\partial\lambda}\Bigr|_{\substack{\lambda=0}}f(\phi(\lambda))$$
Then...
$$\phi(\lambda) = x^{-1}\circ x^\mu(\lambda)$$
$$f(\phi(\lambda)) = f(x^{-1}\circ x^\mu(\lambda)) = (f\circ x^{-1}) \circ (x^\mu(\lambda))$$
$$\frac{\partial}{\partial\lambda}\Bigr|_{\substack{\lambda=0}}(f\circ x^{-1}) \circ (x^\mu(\lambda)) = \frac{\partial(f\circ x^{-1})}{\partial x^\mu}\frac{\partial x^\mu}{\partial\lambda}\Bigr|_{\substack{\lambda=0}}$$ $$(f\circ x^{-1}) = f(x^\mu)$$ $$V_\phi(f) = \frac{\partial f}{\partial x^\mu}\frac{\partial x^\mu}{\partial\lambda}\Bigr|_{\substack{\lambda=0}} = (\frac{\partial x^\mu}{\partial\lambda}\Bigr|_{\substack{\lambda=0}} \frac{\partial}{\partial x^\mu}) f $$
$$\Rightarrow \frac{\partial x^\mu}{\partial\lambda}\Bigr|_{\substack{\lambda=0}} \frac{\partial}{\partial x^\mu} = V^\mu\partial_\mu$$
What is an example of something that the basis $\partial_\mu$ would act on. Also arent we getting tangent vectors to the curve from just the parameter derivative, why do we need this derivative of $f$, what is $f$ actually meant to represent, just some random scalar field? I think my understanding is very flawed.