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