Answer:
[tex]\left \{ {{f(x)=2;\text{ For }0\leq x<1}\\\\\\\\ \atop {f(x)=-\sqrt{x-1}+1;\text{ For }x>1}} \right.[/tex]
Step-by-step explanation:
We have been given graph of a piece-wise function. We are asked to write the equation for the given piece-wise function.
We can see that for x values greater than or equal to 0 and less than 1, the function is a line.
We can see that our given line is a horizontal line, so it will be in form [tex]y=C[/tex] intersecting y-axis at [tex](0,C)[/tex].
The given crosses y-axis at (0,2), therefore, its equation would be [tex]y=2[/tex].
We can see that for x values greater than 1, the graph is a reflected and shifted square root function.
The square root function is reflected across x-axis and has a vertex at (1,1), therefore, our function for second part would be [tex]y=-\sqrt{x-1}+1[/tex].
The function is not defined at [tex]x=1[/tex] and it can a jump continuity.
Therefore, our required piece-wise function would be:
[tex]\left \{ {{f(x)=2;\text{ For }0\leq x<1}\\\\\\\\ \atop {f(x)=-\sqrt{x-1}+1;\text{ For }x>1}} \right.[/tex]
