Answer:
The required piece-wise function is
[tex]f(x)=\begin{cases}x^3-3 & \text{ if } x\leq 2 \\ x^2+6 & \text{ if } x>2 \end{cases}[/tex]
Step-by-step explanation:
From the given graph it is clear that it is a piece-wise function because it is braked in two parts.
One part is defined for x≤2 and other part is defined for x>2 because left curve has closed circle at x=2 and right curve has open circle at x=2.
The y-intercept of the function is -3. From the given graph it is clear that the graph defined for x≤2 is a graph of cubic function that shifts 3 units down. So, the required function for x≤2.
[tex]f(x)=x^3-3[/tex]
From the given graph it is clear that the right curve passes through the point (3,15) and moves towards (2,10).
The function [tex]f(x)=x^2+6[/tex] satisfied by both the points. So, the required function for x>2.
[tex]f(x)=x^2+6[/tex]
Therefore the required piece-wise function is
[tex]f(x)=\begin{cases}x^3-3 & \text{ if } x\leq 2 \\ x^2+6 & \text{ if } x>2 \end{cases}[/tex]
