# Distance between two functions must be greater than 1

The functions used for this problem are simplified functions:

I have a function $$g(x)=x_1^2$$ and I have a function $$h(x,b)=x+b$$ and the Area (let's say in the interval x=[0,5]) between these two functions should be minimized while

constraint: the line represented by the function $$h(x)$$ must always be a minimum of $$1$$ unit under/away from g(x).

So looking at the point x=[0,0], the value $$b$$ in function $$h(x)$$ must be smaller or equal to $$-1$$.

In now need to setup the constraint as a mathematical inequation $$j(x,b)\leqq1$$ but I don't really know how this is done. I only know that the function $$j(x,y)$$ needs to calculate the distance between the two functions and I already read on the internet about it but the people suggested that the distance is calculated like this:

$$j(x,b)=|g(x)-h(x)|$$

That doesn't really make sense for me since in this equation only the vertical distance is being considered. Does someone have any other suggestions (or clarifications)?

• When you say "the line represented by the function h(x) must always be a minimum of 1 unit under/away from g(x)", do you mean pointwise distance or is there a way of measuring distance between two functions? – Aniruddha Deshmukh Apr 21 at 12:50
• From the description you have given, $j \left( x, b \right) \geq 1$. – Aniruddha Deshmukh Apr 21 at 12:50
• @AniruddhaDeshmukh I ment pointwise. Yes, you are right. Has to be "greater than" of course. – Loeli Apr 21 at 13:14
• You use both "under" and "away" to describe the distance between the curves $y=g(x)$ and $y=h(x)$. But "under" specifically refers to vertical distance, so it is hardly surprising that people give you a vertical distance formula. And even more so, as you describe it as a distance between "functions", which is also measured vertically, instead of a distance between "curves". "Away" is less specific in its meaning, so adding it does not clarify the situation. – Paul Sinclair Apr 21 at 20:58