# Define a function "draw curve" between two points with domain $[0,1]$

This might be a simple question, but I spent long time couldn't figure it out $$\dots$$ Say we have two points in $$\mathbb{R}^2$$ such that $$(a,f(a))$$ and $$(c,f(c))$$, as a example we take $$f(x)=\frac{1}{x}$$ where $$x>0$$. I know how to define a function $$g:[0,1]\to\mathbb{R}^2$$ that its graph draw a line between two points, but how do I let it draw a curve which follows the function $$f$$ between two points ?

Here is a graph from Desmos, the red line is a function with domain $$[0,1]$$ that draw a line between two points, the blue function is which I want to define, however I couldn't transfer its domain to $$[0,1]$$. I also tried to modify the red function, but seems didn't work out very well $$\dots$$

Any help would be appreciated!

$$\begin{array}{crcl} g: & [0, 1] &\to& \mathbb{R}^2 \\ & t &\mapsto & (a + t(c - a), f\left(a + t(c - a)\right)) \end{array}$$
You can actually see that $$g(0) = (a, f(a))$$ and $$g(1) = (c, f(c))$$